ABCF->ab-angle a

Percentage Accurate: 18.9% → 58.2%

Time: 18.2s

Alternatives: 10

Speedup: 16.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.9% 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: 58.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := t\_0 \cdot 2\\ t_2 := \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)\\ t_3 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_4 := C \cdot \left(A \cdot 4\right)\\ t_5 := \left(F \cdot \left({B\_m}^{2} - t\_4\right)\right) \cdot 2\\ t_6 := t\_4 - {B\_m}^{2}\\ t_7 := \frac{\sqrt{\left(\left(C + A\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right) \cdot t\_5}}{t\_6}\\ t_8 := t\_2 \cdot F\\ \mathbf{if}\;t\_7 \leq -\infty:\\ \;\;\;\;\frac{\left(-\sqrt{t\_1}\right) \cdot \sqrt{t\_8}}{t\_3}\\ \mathbf{elif}\;t\_7 \leq -1 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(C + A\right) \cdot \left(A - C\right), \frac{1}{A - C}, \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right) \cdot t\_5}}{t\_6}\\ \mathbf{elif}\;t\_7 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot t\_8}}{-t\_3}\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot t\_0}}{-t\_0} \cdot \sqrt{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (* t_0 2.0))
        (t_2 (fma -0.5 (/ (* B_m B_m) A) (* C 2.0)))
        (t_3 (fma (* C A) -4.0 (* B_m B_m)))
        (t_4 (* C (* A 4.0)))
        (t_5 (* (* F (- (pow B_m 2.0) t_4)) 2.0))
        (t_6 (- t_4 (pow B_m 2.0)))
        (t_7
         (/
          (sqrt (* (+ (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))) t_5))
          t_6))
        (t_8 (* t_2 F)))
   (if (<= t_7 (- INFINITY))
     (/ (* (- (sqrt t_1)) (sqrt t_8)) t_3)
     (if (<= t_7 -1e-176)
       (/
        (sqrt
         (*
          (fma
           (* (+ C A) (- A C))
           (/ 1.0 (- A C))
           (sqrt (fma (- A C) (- A C) (* B_m B_m))))
          t_5))
        t_6)
       (if (<= t_7 4e+164)
         (/ (sqrt (* t_1 t_8)) (- t_3))
         (if (<= t_7 INFINITY)
           (* (/ (sqrt (* (* F 2.0) t_0)) (- t_0)) (sqrt t_2))
           (/ (- (sqrt F)) (sqrt (* 0.5 B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = t_0 * 2.0;
	double t_2 = fma(-0.5, ((B_m * B_m) / A), (C * 2.0));
	double t_3 = fma((C * A), -4.0, (B_m * B_m));
	double t_4 = C * (A * 4.0);
	double t_5 = (F * (pow(B_m, 2.0) - t_4)) * 2.0;
	double t_6 = t_4 - pow(B_m, 2.0);
	double t_7 = sqrt((((C + A) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))) * t_5)) / t_6;
	double t_8 = t_2 * F;
	double tmp;
	if (t_7 <= -((double) INFINITY)) {
		tmp = (-sqrt(t_1) * sqrt(t_8)) / t_3;
	} else if (t_7 <= -1e-176) {
		tmp = sqrt((fma(((C + A) * (A - C)), (1.0 / (A - C)), sqrt(fma((A - C), (A - C), (B_m * B_m)))) * t_5)) / t_6;
	} else if (t_7 <= 4e+164) {
		tmp = sqrt((t_1 * t_8)) / -t_3;
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = (sqrt(((F * 2.0) * t_0)) / -t_0) * sqrt(t_2);
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(t_0 * 2.0)
	t_2 = fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))
	t_3 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_4 = Float64(C * Float64(A * 4.0))
	t_5 = Float64(Float64(F * Float64((B_m ^ 2.0) - t_4)) * 2.0)
	t_6 = Float64(t_4 - (B_m ^ 2.0))
	t_7 = Float64(sqrt(Float64(Float64(Float64(C + A) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))) * t_5)) / t_6)
	t_8 = Float64(t_2 * F)
	tmp = 0.0
	if (t_7 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-sqrt(t_1)) * sqrt(t_8)) / t_3);
	elseif (t_7 <= -1e-176)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(C + A) * Float64(A - C)), Float64(1.0 / Float64(A - C)), sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))) * t_5)) / t_6);
	elseif (t_7 <= 4e+164)
		tmp = Float64(sqrt(Float64(t_1 * t_8)) / Float64(-t_3));
	elseif (t_7 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * 2.0) * t_0)) / Float64(-t_0)) * sqrt(t_2));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$2 * F), $MachinePrecision]}, If[LessEqual[t$95$7, (-Infinity)], N[(N[((-N[Sqrt[t$95$1], $MachinePrecision]) * N[Sqrt[t$95$8], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$7, -1e-176], N[(N[Sqrt[N[(N[(N[(N[(C + A), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(A - C), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$7, 4e+164], N[(N[Sqrt[N[(t$95$1 * t$95$8), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision], If[LessEqual[t$95$7, Infinity], N[(N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.2%

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

   3. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 31.0%

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

   9. Applied rewrites 51.1%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-176

   1. Initial program 97.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. lift--.f64 N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 97.3

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

      15. lift-+.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-fma.f64 97.3

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 97.3

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

   4. Applied rewrites 97.3%

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

   if -1e-176 < (/.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))) < 4e164

   1. Initial program 17.2%

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

   3. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

   9. Applied rewrites 72.3%

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

   if 4e164 < (/.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 28.0%

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

   3. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.5

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

   5. Applied rewrites 56.5%

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

   7. Applied rewrites 56.5%

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

   9. Applied rewrites 84.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 29.0

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

   5. Applied rewrites 29.0%

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

   7. Applied rewrites 29.0%

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

   9. Applied rewrites 29.1%

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

   11. Applied rewrites 41.2%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 60.4%

   \[\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, C \cdot A, B \cdot B\right) \cdot 2}\right) \cdot \sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot F}}{\mathsf{fma}\left(C \cdot A, -4, 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 -1 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(C + A\right) \cdot \left(A - C\right), \frac{1}{A - C}, \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \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 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot F\right)}}{-\mathsf{fma}\left(C \cdot A, -4, 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 \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := t\_0 \cdot 2\\ t_2 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_3 := C \cdot \left(A \cdot 4\right)\\ t_4 := \frac{\sqrt{\left(\left(C + A\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_3\right)\right) \cdot 2\right)}}{t\_3 - {B\_m}^{2}}\\ t_5 := \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)\\ t_6 := t\_5 \cdot F\\ t_7 := -t\_0\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\left(-\sqrt{t\_1}\right) \cdot \sqrt{t\_6}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{\left(\left(t\_0 \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)} + \left(C + A\right)\right)}}{t\_7}\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot t\_6}}{-t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot t\_0}}{t\_7} \cdot \sqrt{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (* t_0 2.0))
        (t_2 (fma (* C A) -4.0 (* B_m B_m)))
        (t_3 (* C (* A 4.0)))
        (t_4
         (/
          (sqrt
           (*
            (+ (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))
            (* (* F (- (pow B_m 2.0) t_3)) 2.0)))
          (- t_3 (pow B_m 2.0))))
        (t_5 (fma -0.5 (/ (* B_m B_m) A) (* C 2.0)))
        (t_6 (* t_5 F))
        (t_7 (- t_0)))
   (if (<= t_4 (- INFINITY))
     (/ (* (- (sqrt t_1)) (sqrt t_6)) t_2)
     (if (<= t_4 -1e-176)
       (/
        (sqrt
         (*
          (* (* t_0 F) 2.0)
          (+ (sqrt (fma (- A C) (- A C) (* B_m B_m))) (+ C A))))
        t_7)
       (if (<= t_4 4e+164)
         (/ (sqrt (* t_1 t_6)) (- t_2))
         (if (<= t_4 INFINITY)
           (* (/ (sqrt (* (* F 2.0) t_0)) t_7) (sqrt t_5))
           (/ (- (sqrt F)) (sqrt (* 0.5 B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = t_0 * 2.0;
	double t_2 = fma((C * A), -4.0, (B_m * B_m));
	double t_3 = C * (A * 4.0);
	double t_4 = sqrt((((C + A) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))) * ((F * (pow(B_m, 2.0) - t_3)) * 2.0))) / (t_3 - pow(B_m, 2.0));
	double t_5 = fma(-0.5, ((B_m * B_m) / A), (C * 2.0));
	double t_6 = t_5 * F;
	double t_7 = -t_0;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (-sqrt(t_1) * sqrt(t_6)) / t_2;
	} else if (t_4 <= -1e-176) {
		tmp = sqrt((((t_0 * F) * 2.0) * (sqrt(fma((A - C), (A - C), (B_m * B_m))) + (C + A)))) / t_7;
	} else if (t_4 <= 4e+164) {
		tmp = sqrt((t_1 * t_6)) / -t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt(((F * 2.0) * t_0)) / t_7) * sqrt(t_5);
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(t_0 * 2.0)
	t_2 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_3 = Float64(C * Float64(A * 4.0))
	t_4 = Float64(sqrt(Float64(Float64(Float64(C + A) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_3)) * 2.0))) / Float64(t_3 - (B_m ^ 2.0)))
	t_5 = fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))
	t_6 = Float64(t_5 * F)
	t_7 = Float64(-t_0)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-sqrt(t_1)) * sqrt(t_6)) / t_2);
	elseif (t_4 <= -1e-176)
		tmp = Float64(sqrt(Float64(Float64(Float64(t_0 * F) * 2.0) * Float64(sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))) + Float64(C + A)))) / t_7);
	elseif (t_4 <= 4e+164)
		tmp = Float64(sqrt(Float64(t_1 * t_6)) / Float64(-t_2));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * 2.0) * t_0)) / t_7) * sqrt(t_5));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(C * N[(A * 4.0), $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$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * F), $MachinePrecision]}, Block[{t$95$7 = (-t$95$0)}, If[LessEqual[t$95$4, (-Infinity)], N[(N[((-N[Sqrt[t$95$1], $MachinePrecision]) * N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, -1e-176], N[(N[Sqrt[N[(N[(N[(t$95$0 * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$7), $MachinePrecision], If[LessEqual[t$95$4, 4e+164], N[(N[Sqrt[N[(t$95$1 * t$95$6), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$7), $MachinePrecision] * N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.2%

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

   3. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 35.1

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

   5. Applied rewrites 35.1%

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

   7. Applied rewrites 35.1%

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

   9. Applied rewrites 48.7%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-176

   1. Initial program 97.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   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 97.7%

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

   if -1e-176 < (/.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))) < 4e164

   1. Initial program 21.3%

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

   3. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

   9. Applied rewrites 59.9%

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

   if 4e164 < (/.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 15.5%

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

   3. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 43.8

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

   5. Applied rewrites 43.8%

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

   7. Applied rewrites 43.8%

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

   9. Applied rewrites 73.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 31.1

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

   5. Applied rewrites 31.1%

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

   7. Applied rewrites 31.2%

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

   9. Applied rewrites 31.2%

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

   11. Applied rewrites 46.9%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 58.2%

   \[\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, C \cdot A, B \cdot B\right) \cdot 2}\right) \cdot \sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot F}}{\mathsf{fma}\left(C \cdot A, -4, 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 -1 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} + \left(C + A\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot 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 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot F\right)}}{-\mathsf{fma}\left(C \cdot A, -4, 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 \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))