ABCF->ab-angle b

Percentage Accurate: 19.1% → 39.4%

Time: 18.5s

Alternatives: 15

Speedup: 7.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 39.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \frac{B \cdot B}{C}\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \left(F \cdot \left({B}^{2} - t\_1\right)\right) \cdot 2\\ t_3 := t\_1 - {B}^{2}\\ t_4 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot t\_2}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(C - A\right) \cdot \left(C + A\right), \frac{-1}{A - C}, -\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \mathsf{fma}\left(\left(\left(C \cdot A\right) \cdot -4\right) \cdot F, 2, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(t\_0, -0.5, A\right) + A\right) \cdot t\_2}}{-\left(t\_0 - A \cdot 4\right) \cdot C}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (/ (* B B) C))
        (t_1 (* C (* A 4.0)))
        (t_2 (* (* F (- (pow B 2.0) t_1)) 2.0))
        (t_3 (- t_1 (pow B 2.0)))
        (t_4
         (/
          (sqrt (* (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))) t_2))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ (* (sqrt (* (+ A A) F)) (sqrt (* (fma -4.0 (* C A) (* B B)) 2.0))) t_3)
     (if (<= t_4 -2e-223)
       (/
        (sqrt
         (*
          (fma
           (* (- C A) (+ C A))
           (/ -1.0 (- A C))
           (- (sqrt (fma (- A C) (- A C) (* B B)))))
          (fma (* (* (* C A) -4.0) F) 2.0 (* (* (* F 2.0) B) B))))
        t_3)
       (/
        (sqrt (* (+ (fma t_0 -0.5 A) A) t_2))
        (- (* (- t_0 (* A 4.0)) C)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) / C;
	double t_1 = C * (A * 4.0);
	double t_2 = (F * (pow(B, 2.0) - t_1)) * 2.0;
	double t_3 = t_1 - pow(B, 2.0);
	double t_4 = sqrt((((C + A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))) * t_2)) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt(((A + A) * F)) * sqrt((fma(-4.0, (C * A), (B * B)) * 2.0))) / t_3;
	} else if (t_4 <= -2e-223) {
		tmp = sqrt((fma(((C - A) * (C + A)), (-1.0 / (A - C)), -sqrt(fma((A - C), (A - C), (B * B)))) * fma((((C * A) * -4.0) * F), 2.0, (((F * 2.0) * B) * B)))) / t_3;
	} else {
		tmp = sqrt(((fma(t_0, -0.5, A) + A) * t_2)) / -((t_0 - (A * 4.0)) * C);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) / C)
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = Float64(Float64(F * Float64((B ^ 2.0) - t_1)) * 2.0)
	t_3 = Float64(t_1 - (B ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))) * t_2)) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(A + A) * F)) * sqrt(Float64(fma(-4.0, Float64(C * A), Float64(B * B)) * 2.0))) / t_3);
	elseif (t_4 <= -2e-223)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(C - A) * Float64(C + A)), Float64(-1.0 / Float64(A - C)), Float64(-sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))) * fma(Float64(Float64(Float64(C * A) * -4.0) * F), 2.0, Float64(Float64(Float64(F * 2.0) * B) * B)))) / t_3);
	else
		tmp = Float64(sqrt(Float64(Float64(fma(t_0, -0.5, A) + A) * t_2)) / Float64(-Float64(Float64(t_0 - Float64(A * 4.0)) * C)));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -2e-223], N[(N[Sqrt[N[(N[(N[(N[(C - A), $MachinePrecision] * N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(A - C), $MachinePrecision]), $MachinePrecision] + (-N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * N[(N[(N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision] * F), $MachinePrecision] * 2.0 + N[(N[(N[(F * 2.0), $MachinePrecision] * B), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(t$95$0 * -0.5 + A), $MachinePrecision] + A), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / (-N[(N[(t$95$0 - N[(A * 4.0), $MachinePrecision]), $MachinePrecision] * C), $MachinePrecision])), $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-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 26.1%

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

   5. Taylor expanded in C around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 25.8

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

   7. Applied rewrites 25.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999999e-223

   1. Initial program 98.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. difference-of-squares N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-neg.f64 98.2

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

      19. lift-+.f64 N/A

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

      20. lift-pow.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-fma.f64 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. distribute-rgt-in N/A

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

      19. *-commutative N/A

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

   6. Applied rewrites 98.4%

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

   if -1.9999999999999999e-223 < (/.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 6.0%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 12.7

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

   5. Applied rewrites 12.7%

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

   6. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 12.7

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

   8. Applied rewrites 12.7%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(C - A\right) \cdot \left(C + A\right), \frac{-1}{A - C}, -\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \mathsf{fma}\left(\left(\left(C \cdot A\right) \cdot -4\right) \cdot F, 2, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{-\left(\frac{B \cdot B}{C} - A \cdot 4\right) \cdot C}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 39.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \left(F \cdot \left({B}^{2} - t\_1\right)\right) \cdot 2\\ t_3 := \frac{B \cdot B}{C}\\ t_4 := t\_1 - {B}^{2}\\ t_5 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot t\_2}}{t\_4}\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \sqrt{t\_0 \cdot 2}}{t\_4}\\ \mathbf{elif}\;t\_5 \leq -2 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_0\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(t\_3, -0.5, A\right) + A\right) \cdot t\_2}}{-\left(t\_3 - A \cdot 4\right) \cdot C}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B B)))
        (t_1 (* C (* A 4.0)))
        (t_2 (* (* F (- (pow B 2.0) t_1)) 2.0))
        (t_3 (/ (* B B) C))
        (t_4 (- t_1 (pow B 2.0)))
        (t_5
         (/
          (sqrt (* (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))) t_2))
          t_4)))
   (if (<= t_5 (- INFINITY))
     (/ (* (sqrt (* (+ A A) F)) (sqrt (* t_0 2.0))) t_4)
     (if (<= t_5 -2e-223)
       (/
        (sqrt
         (*
          (* (* F 2.0) t_0)
          (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B))))))
        (- t_0))
       (/
        (sqrt (* (+ (fma t_3 -0.5 A) A) t_2))
        (- (* (- t_3 (* A 4.0)) C)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B * B));
	double t_1 = C * (A * 4.0);
	double t_2 = (F * (pow(B, 2.0) - t_1)) * 2.0;
	double t_3 = (B * B) / C;
	double t_4 = t_1 - pow(B, 2.0);
	double t_5 = sqrt((((C + A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))) * t_2)) / t_4;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = (sqrt(((A + A) * F)) * sqrt((t_0 * 2.0))) / t_4;
	} else if (t_5 <= -2e-223) {
		tmp = sqrt((((F * 2.0) * t_0) * ((C + A) - sqrt(fma((A - C), (A - C), (B * B)))))) / -t_0;
	} else {
		tmp = sqrt(((fma(t_3, -0.5, A) + A) * t_2)) / -((t_3 - (A * 4.0)) * C);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B * B))
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = Float64(Float64(F * Float64((B ^ 2.0) - t_1)) * 2.0)
	t_3 = Float64(Float64(B * B) / C)
	t_4 = Float64(t_1 - (B ^ 2.0))
	t_5 = Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))) * t_2)) / t_4)
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(A + A) * F)) * sqrt(Float64(t_0 * 2.0))) / t_4);
	elseif (t_5 <= -2e-223)
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_0) * Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(fma(t_3, -0.5, A) + A) * t_2)) / Float64(-Float64(Float64(t_3 - Float64(A * 4.0)) * C)));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$5, -2e-223], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(t$95$3 * -0.5 + A), $MachinePrecision] + A), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / (-N[(N[(t$95$3 - N[(A * 4.0), $MachinePrecision]), $MachinePrecision] * C), $MachinePrecision])), $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-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 26.1%

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

   5. Taylor expanded in C around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 25.8

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

   7. Applied rewrites 25.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999999e-223

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      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 98.2%

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

   if -1.9999999999999999e-223 < (/.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 6.0%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 12.7

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

   5. Applied rewrites 12.7%

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

   6. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 12.7

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

   8. Applied rewrites 12.7%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{-\left(\frac{B \cdot B}{C} - A \cdot 4\right) \cdot C}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 39.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_3 := t\_1 - {B}^{2}\\ t_4 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \sqrt{t\_0 \cdot 2}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_0\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(t\_2 \cdot \left(F \cdot 2\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)}}{-t\_2}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B B)))
        (t_1 (* C (* A 4.0)))
        (t_2 (fma (* -4.0 C) A (* B B)))
        (t_3 (- t_1 (pow B 2.0)))
        (t_4
         (/
          (sqrt
           (*
            (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))
            (* (* F (- (pow B 2.0) t_1)) 2.0)))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ (* (sqrt (* (+ A A) F)) (sqrt (* t_0 2.0))) t_3)
     (if (<= t_4 -2e-223)
       (/
        (sqrt
         (*
          (* (* F 2.0) t_0)
          (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B))))))
        (- t_0))
       (/
        (sqrt (* (* t_2 (* F 2.0)) (+ (fma -0.5 (/ (* B B) C) A) A)))
        (- t_2))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B * B));
	double t_1 = C * (A * 4.0);
	double t_2 = fma((-4.0 * C), A, (B * B));
	double t_3 = t_1 - pow(B, 2.0);
	double t_4 = sqrt((((C + A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))) * ((F * (pow(B, 2.0) - t_1)) * 2.0))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt(((A + A) * F)) * sqrt((t_0 * 2.0))) / t_3;
	} else if (t_4 <= -2e-223) {
		tmp = sqrt((((F * 2.0) * t_0) * ((C + A) - sqrt(fma((A - C), (A - C), (B * B)))))) / -t_0;
	} else {
		tmp = sqrt(((t_2 * (F * 2.0)) * (fma(-0.5, ((B * B) / C), A) + A))) / -t_2;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B * B))
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = fma(Float64(-4.0 * C), A, Float64(B * B))
	t_3 = Float64(t_1 - (B ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))) * Float64(Float64(F * Float64((B ^ 2.0) - t_1)) * 2.0))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(A + A) * F)) * sqrt(Float64(t_0 * 2.0))) / t_3);
	elseif (t_4 <= -2e-223)
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_0) * Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(t_2 * Float64(F * 2.0)) * Float64(fma(-0.5, Float64(Float64(B * B) / C), A) + A))) / Float64(-t_2));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -2e-223], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(N[(t$95$2 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision] + A), $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))) < -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-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 26.1%

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

   5. Taylor expanded in C around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 25.8

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

   7. Applied rewrites 25.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999999e-223

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      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 98.2%

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

   if -1.9999999999999999e-223 < (/.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 6.0%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 12.7

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

   5. Applied rewrites 12.7%

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

   7. Applied rewrites 12.7%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_2 - {B}^{2}}\\ t_4 := \frac{B \cdot B}{C}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\left(-\sqrt{t\_0 \cdot 2}\right) \cdot \sqrt{\left(\mathsf{fma}\left(t\_4, -0.5, A\right) + A\right) \cdot F}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_1\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(F \cdot 2\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, t\_4, A\right) + A\right)}}{-t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B B)))
        (t_1 (fma -4.0 (* C A) (* B B)))
        (t_2 (* C (* A 4.0)))
        (t_3
         (/
          (sqrt
           (*
            (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))
            (* (* F (- (pow B 2.0) t_2)) 2.0)))
          (- t_2 (pow B 2.0))))
        (t_4 (/ (* B B) C)))
   (if (<= t_3 (- INFINITY))
     (/ (* (- (sqrt (* t_0 2.0))) (sqrt (* (+ (fma t_4 -0.5 A) A) F))) t_0)
     (if (<= t_3 -2e-223)
       (/
        (sqrt
         (*
          (* (* F 2.0) t_1)
          (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B))))))
        (- t_1))
       (/ (sqrt (* (* t_0 (* F 2.0)) (+ (fma -0.5 t_4 A) A))) (- t_0))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B * B));
	double t_1 = fma(-4.0, (C * A), (B * B));
	double t_2 = C * (A * 4.0);
	double t_3 = sqrt((((C + A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))) * ((F * (pow(B, 2.0) - t_2)) * 2.0))) / (t_2 - pow(B, 2.0));
	double t_4 = (B * B) / C;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (-sqrt((t_0 * 2.0)) * sqrt(((fma(t_4, -0.5, A) + A) * F))) / t_0;
	} else if (t_3 <= -2e-223) {
		tmp = sqrt((((F * 2.0) * t_1) * ((C + A) - sqrt(fma((A - C), (A - C), (B * B)))))) / -t_1;
	} else {
		tmp = sqrt(((t_0 * (F * 2.0)) * (fma(-0.5, t_4, A) + A))) / -t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B * B))
	t_1 = fma(-4.0, Float64(C * A), Float64(B * B))
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))) * Float64(Float64(F * Float64((B ^ 2.0) - t_2)) * 2.0))) / Float64(t_2 - (B ^ 2.0)))
	t_4 = Float64(Float64(B * B) / C)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-sqrt(Float64(t_0 * 2.0))) * sqrt(Float64(Float64(fma(t_4, -0.5, A) + A) * F))) / t_0);
	elseif (t_3 <= -2e-223)
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_1) * Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))))) / Float64(-t_1));
	else
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(F * 2.0)) * Float64(fma(-0.5, t_4, A) + A))) / Float64(-t_0));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[((-N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[N[(N[(N[(t$95$4 * -0.5 + A), $MachinePrecision] + A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$3, -2e-223], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(N[Sqrt[N[(N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * t$95$4 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $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. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 21.3

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

   5. Applied rewrites 21.3%

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

   7. Applied rewrites 21.3%

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

   9. Applied rewrites 27.4%

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

   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))) < -1.9999999999999999e-223

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      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 98.2%

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

   if -1.9999999999999999e-223 < (/.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 6.0%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 12.7

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

   5. Applied rewrites 12.7%

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

   7. Applied rewrites 12.7%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\left(-\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2}\right) \cdot \sqrt{\left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right) \cdot F}}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(t\_0 \cdot \left(F \cdot 2\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B B))))
   (if (<= (pow B 2.0) 2e-23)
     (/ (sqrt (* (+ A A) (* t_0 (* F 2.0)))) (- t_0))
     (*
      (sqrt
       (*
        (/
         (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B))))
         (fma (* -4.0 A) C (* B B)))
        F))
      (- (sqrt 2.0))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B * B));
	double tmp;
	if (pow(B, 2.0) <= 2e-23) {
		tmp = sqrt(((A + A) * (t_0 * (F * 2.0)))) / -t_0;
	} else {
		tmp = sqrt(((((C + A) - sqrt(fma((A - C), (A - C), (B * B)))) / fma((-4.0 * A), C, (B * B))) * F)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B * B))
	tmp = 0.0
	if ((B ^ 2.0) <= 2e-23)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(t_0 * Float64(F * 2.0)))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))) / fma(Float64(-4.0 * A), C, Float64(B * B))) * F)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-23], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999992e-23

   1. Initial program 21.5%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 21.1

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

   5. Applied rewrites 21.1%

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

   7. Applied rewrites 21.1%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 20.1

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

   10. Applied rewrites 20.1%

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

   if 1.99999999999999992e-23 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 18.2%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 28.8%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 27.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;{B}^{2} \leq 10^{-49}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(t\_0 \cdot \left(F \cdot 2\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{-1}{\left(\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)} - A\right) \cdot F}} \cdot \frac{B}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B B))))
   (if (<= (pow B 2.0) 1e-49)
     (/ (sqrt (* (+ A A) (* t_0 (* F 2.0)))) (- t_0))
     (/
      -1.0
      (*
       (sqrt (/ -1.0 (* (- (sqrt (fma A A (* B B))) A) F)))
       (/ B (sqrt 2.0)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B * B));
	double tmp;
	if (pow(B, 2.0) <= 1e-49) {
		tmp = sqrt(((A + A) * (t_0 * (F * 2.0)))) / -t_0;
	} else {
		tmp = -1.0 / (sqrt((-1.0 / ((sqrt(fma(A, A, (B * B))) - A) * F))) * (B / sqrt(2.0)));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B * B))
	tmp = 0.0
	if ((B ^ 2.0) <= 1e-49)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(t_0 * Float64(F * 2.0)))) / Float64(-t_0));
	else
		tmp = Float64(-1.0 / Float64(sqrt(Float64(-1.0 / Float64(Float64(sqrt(fma(A, A, Float64(B * B))) - A) * F))) * Float64(B / sqrt(2.0))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e-49], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[N[(-1.0 / N[(N[(N[Sqrt[N[(A * A + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - A), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(B / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999936e-50

   1. Initial program 21.7%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 21.3

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

   5. Applied rewrites 21.3%

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

   7. Applied rewrites 21.3%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 21.1

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

   10. Applied rewrites 21.1%

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

   if 9.99999999999999936e-50 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 18.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 18.1%

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

   5. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 11.1

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

   7. Applied rewrites 11.1%

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

4. Final simplification 16.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-49}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{-1}{\left(\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)} - A\right) \cdot F}} \cdot \frac{B}{\sqrt{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 27.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;{B}^{2} \leq 10^{-49}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(t\_0 \cdot \left(F \cdot 2\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B B))))
   (if (<= (pow B 2.0) 1e-49)
     (/ (sqrt (* (+ A A) (* t_0 (* F 2.0)))) (- t_0))
     (* (sqrt (* (- A (sqrt (fma B B (* A A)))) F)) (/ (- (sqrt 2.0)) B)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B * B));
	double tmp;
	if (pow(B, 2.0) <= 1e-49) {
		tmp = sqrt(((A + A) * (t_0 * (F * 2.0)))) / -t_0;
	} else {
		tmp = sqrt(((A - sqrt(fma(B, B, (A * A)))) * F)) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B * B))
	tmp = 0.0
	if ((B ^ 2.0) <= 1e-49)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(t_0 * Float64(F * 2.0)))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(A - sqrt(fma(B, B, Float64(A * A)))) * F)) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e-49], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999936e-50

   1. Initial program 21.7%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 21.3

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

   5. Applied rewrites 21.3%

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

   7. Applied rewrites 21.3%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 21.1

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

   10. Applied rewrites 21.1%

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

   if 9.99999999999999936e-50 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 18.1%

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

   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

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 11.2

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

   5. Applied rewrites 11.2%

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

4. Final simplification 15.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-49}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 17.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t\_0} \cdot \sqrt{\left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B B))))
   (if (<= (pow B 2.0) 2e-23)
     (/ (sqrt (* (* (* (* A A) C) F) -16.0)) (- t_0))
     (* (/ -1.0 t_0) (sqrt (* (* (* (* B B) B) F) -2.0))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B * B));
	double tmp;
	if (pow(B, 2.0) <= 2e-23) {
		tmp = sqrt(((((A * A) * C) * F) * -16.0)) / -t_0;
	} else {
		tmp = (-1.0 / t_0) * sqrt(((((B * B) * B) * F) * -2.0));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B * B))
	tmp = 0.0
	if ((B ^ 2.0) <= 2e-23)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(A * A) * C) * F) * -16.0)) / Float64(-t_0));
	else
		tmp = Float64(Float64(-1.0 / t_0) * sqrt(Float64(Float64(Float64(Float64(B * B) * B) * F) * -2.0)));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-23], N[(N[Sqrt[N[(N[(N[(N[(A * A), $MachinePrecision] * C), $MachinePrecision] * F), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(B * B), $MachinePrecision] * B), $MachinePrecision] * F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999992e-23

   1. Initial program 21.5%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 21.1

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

   5. Applied rewrites 21.1%

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

   7. Applied rewrites 21.1%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 13.6

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

   10. Applied rewrites 13.6%

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

   if 1.99999999999999992e-23 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 18.2%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 7.1

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

   5. Applied rewrites 7.1%

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

   7. Applied rewrites 7.1%

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

   8. Taylor expanded in B around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left({B}^{3} \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 6.3

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

   10. Applied rewrites 6.3%

       \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{\left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right) \cdot -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 17.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := -\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right) \cdot -2}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (fma (* -4.0 C) A (* B B)))))
   (if (<= (pow B 2.0) 2e-23)
     (/ (sqrt (* (* (* (* A A) C) F) -16.0)) t_0)
     (/ (sqrt (* (* (* (* B B) B) F) -2.0)) t_0))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = -fma((-4.0 * C), A, (B * B));
	double tmp;
	if (pow(B, 2.0) <= 2e-23) {
		tmp = sqrt(((((A * A) * C) * F) * -16.0)) / t_0;
	} else {
		tmp = sqrt(((((B * B) * B) * F) * -2.0)) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(-fma(Float64(-4.0 * C), A, Float64(B * B)))
	tmp = 0.0
	if ((B ^ 2.0) <= 2e-23)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(A * A) * C) * F) * -16.0)) / t_0);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(B * B) * B) * F) * -2.0)) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = (-N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-23], N[(N[Sqrt[N[(N[(N[(N[(A * A), $MachinePrecision] * C), $MachinePrecision] * F), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(B * B), $MachinePrecision] * B), $MachinePrecision] * F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999992e-23

   1. Initial program 21.5%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 21.1

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

   5. Applied rewrites 21.1%

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

   7. Applied rewrites 21.1%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 13.6

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

   10. Applied rewrites 13.6%

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

   if 1.99999999999999992e-23 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 18.2%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 7.1

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

   5. Applied rewrites 7.1%

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

   7. Applied rewrites 7.1%

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

   8. Taylor expanded in B around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \color{blue}{\left({B}^{3} \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 6.3

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

   10. Applied rewrites 6.3%

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

4. Final simplification 10.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right) \cdot -2}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 3 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 3e-24)
   (/ (sqrt (* (* (* (* (+ A A) F) C) A) -8.0)) (- (fma (* -4.0 C) A (* B B))))
   (* (sqrt (* (- A (sqrt (fma B B (* A A)))) F)) (/ (- (sqrt 2.0)) B))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 3e-24) {
		tmp = sqrt((((((A + A) * F) * C) * A) * -8.0)) / -fma((-4.0 * C), A, (B * B));
	} else {
		tmp = sqrt(((A - sqrt(fma(B, B, (A * A)))) * F)) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 3e-24)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(A + A) * F) * C) * A) * -8.0)) / Float64(-fma(Float64(-4.0 * C), A, Float64(B * B))));
	else
		tmp = Float64(sqrt(Float64(Float64(A - sqrt(fma(B, B, Float64(A * A)))) * F)) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 3e-24], N[(N[Sqrt[N[(N[(N[(N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision] * C), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.99999999999999995e-24

   1. Initial program 21.9%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 16.9%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 14.4

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

   10. Applied rewrites 14.4%

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

   if 2.99999999999999995e-24 < B

   1. Initial program 14.2%

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

   3. Taylor expanded in 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

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 20.3

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

   5. Applied rewrites 20.3%

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

4. Final simplification 16.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 24.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;B \leq 3.1 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t\_0} \cdot \sqrt{\left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B B))))
   (if (<= B 3.1e-24)
     (/ (sqrt (* (* (* (* (+ A A) F) C) A) -8.0)) (- t_0))
     (* (/ -1.0 t_0) (sqrt (* (* (* (* B B) B) F) -2.0))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B * B));
	double tmp;
	if (B <= 3.1e-24) {
		tmp = sqrt((((((A + A) * F) * C) * A) * -8.0)) / -t_0;
	} else {
		tmp = (-1.0 / t_0) * sqrt(((((B * B) * B) * F) * -2.0));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B * B))
	tmp = 0.0
	if (B <= 3.1e-24)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(A + A) * F) * C) * A) * -8.0)) / Float64(-t_0));
	else
		tmp = Float64(Float64(-1.0 / t_0) * sqrt(Float64(Float64(Float64(Float64(B * B) * B) * F) * -2.0)));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 3.1e-24], N[(N[Sqrt[N[(N[(N[(N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision] * C), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(B * B), $MachinePrecision] * B), $MachinePrecision] * F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 3.1e-24

   1. Initial program 21.9%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 16.9%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 14.4

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

   10. Applied rewrites 14.4%

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

   if 3.1e-24 < B

   1. Initial program 14.2%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 6.7

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

   5. Applied rewrites 6.7%

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

   7. Applied rewrites 6.7%

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

   8. Taylor expanded in B around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left({B}^{3} \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 12.2

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

   10. Applied rewrites 12.2%

       \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.1 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{\left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right) \cdot -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 17.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \frac{\sqrt{\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (/ (sqrt (* (* (* (* A A) C) F) -16.0)) (- (fma (* -4.0 C) A (* B B)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return sqrt(((((A * A) * C) * F) * -16.0)) / -fma((-4.0 * C), A, (B * B));
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(sqrt(Float64(Float64(Float64(Float64(A * A) * C) * F) * -16.0)) / Float64(-fma(Float64(-4.0 * C), A, Float64(B * B))))
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(N[Sqrt[N[(N[(N[(N[(A * A), $MachinePrecision] * C), $MachinePrecision] * F), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in C around inf

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 14.2

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

5. Applied rewrites 14.2%

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

7. Applied rewrites 14.2%

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

8. Taylor expanded in A around -inf

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

   1. lower-*.f64 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 8.3

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

10. Applied rewrites 8.3%

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

11. Final simplification 8.3%

    \[\leadsto \frac{\sqrt{\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
12. Add Preprocessing

Alternative 13: 2.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (/ (sqrt (* (* (* (* C C) F) A) -16.0)) (- (fma (* -4.0 C) A (* B B)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return sqrt(((((C * C) * F) * A) * -16.0)) / -fma((-4.0 * C), A, (B * B));
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(sqrt(Float64(Float64(Float64(Float64(C * C) * F) * A) * -16.0)) / Float64(-fma(Float64(-4.0 * C), A, Float64(B * B))))
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision] * A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in C around inf

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 14.2

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

5. Applied rewrites 14.2%

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

7. Applied rewrites 14.2%

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

8. Taylor expanded in C around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 11.3

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

10. Applied rewrites 11.3%

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

11. Final simplification 11.3%

    \[\leadsto \frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
12. Add Preprocessing

Alternative 14: 2.0% accurate, 14.9× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \sqrt{\frac{2}{\frac{B}{F}}} \end{array} \]

FPCore

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

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return sqrt((2.0 / (B / F)));
}

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
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
    code = sqrt((2.0d0 / (b / f)))
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return Math.sqrt((2.0 / (B / F)));
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return math.sqrt((2.0 / (B / F)))

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return sqrt(Float64(2.0 / Float64(B / F)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in B around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
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. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 2.0

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

5. Applied rewrites 2.0%

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

7. Applied rewrites 2.0%

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

9. Applied rewrites 2.0%

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

Alternative 15: 2.0% accurate, 18.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \sqrt{\frac{2}{B} \cdot F} \end{array} \]

FPCore

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

C

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

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
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
    code = sqrt(((2.0d0 / b) * f))
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return Math.sqrt(((2.0 / B) * F));
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return math.sqrt(((2.0 / B) * F))

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return sqrt(Float64(Float64(2.0 / B) * F))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in B around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
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. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 2.0

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

5. Applied rewrites 2.0%

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

7. Applied rewrites 2.0%

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

9. Applied rewrites 2.0%

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

10. Final simplification 2.0%

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

Reproduce

herbie shell --seed 2024235 
(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))))