ABCF->ab-angle a

Percentage Accurate: 18.9% → 60.5%

Time: 20.1s

Alternatives: 18

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: 60.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \sqrt{2 \cdot \left(2 \cdot C\right)}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := 2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\\ t_4 := t\_2 - {B\_m}^{2}\\ t_5 := \frac{\sqrt{t\_3 \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_4}\\ t_6 := \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{t\_0} \cdot \left(\sqrt{F} \cdot t\_1\right)}{t\_4}\\ \mathbf{elif}\;t\_5 \leq -1 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot \mathsf{fma}\left(\left(A + C\right) \cdot \left(A - C\right), \frac{1}{A - C}, \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{t\_4}\\ \mathbf{elif}\;t\_5 \leq 10^{+225}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot t\_6\right) \cdot \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, 2 \cdot C\right)\right)}}{-t\_6}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\sqrt{F \cdot t\_0} \cdot \left(t\_1 \cdot \frac{-1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (sqrt (* 2.0 (* 2.0 C))))
        (t_2 (* (* 4.0 A) C))
        (t_3 (* 2.0 (* (- (pow B_m 2.0) t_2) F)))
        (t_4 (- t_2 (pow B_m 2.0)))
        (t_5
         (/
          (sqrt (* t_3 (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_4))
        (t_6 (fma C (* A -4.0) (* B_m B_m))))
   (if (<= t_5 (- INFINITY))
     (/ (* (sqrt t_0) (* (sqrt F) t_1)) t_4)
     (if (<= t_5 -1e-176)
       (/
        (sqrt
         (*
          t_3
          (fma
           (* (+ A C) (- A C))
           (/ 1.0 (- A C))
           (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        t_4)
       (if (<= t_5 1e+225)
         (/
          (sqrt (* 2.0 (* (* F t_6) (fma (/ (* B_m B_m) A) -0.5 (* 2.0 C)))))
          (- t_6))
         (if (<= t_5 INFINITY)
           (* (sqrt (* F t_0)) (* t_1 (/ -1.0 t_0)))
           (- (/ (sqrt F) (sqrt (* B_m 0.5))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = sqrt((2.0 * (2.0 * C)));
	double t_2 = (4.0 * A) * C;
	double t_3 = 2.0 * ((pow(B_m, 2.0) - t_2) * F);
	double t_4 = t_2 - pow(B_m, 2.0);
	double t_5 = sqrt((t_3 * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_4;
	double t_6 = fma(C, (A * -4.0), (B_m * B_m));
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = (sqrt(t_0) * (sqrt(F) * t_1)) / t_4;
	} else if (t_5 <= -1e-176) {
		tmp = sqrt((t_3 * fma(((A + C) * (A - C)), (1.0 / (A - C)), sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / t_4;
	} else if (t_5 <= 1e+225) {
		tmp = sqrt((2.0 * ((F * t_6) * fma(((B_m * B_m) / A), -0.5, (2.0 * C))))) / -t_6;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = sqrt((F * t_0)) * (t_1 * (-1.0 / t_0));
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = sqrt(Float64(2.0 * Float64(2.0 * C)))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F))
	t_4 = Float64(t_2 - (B_m ^ 2.0))
	t_5 = Float64(sqrt(Float64(t_3 * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_4)
	t_6 = fma(C, Float64(A * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(t_0) * Float64(sqrt(F) * t_1)) / t_4);
	elseif (t_5 <= -1e-176)
		tmp = Float64(sqrt(Float64(t_3 * fma(Float64(Float64(A + C) * Float64(A - C)), Float64(1.0 / Float64(A - C)), sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / t_4);
	elseif (t_5 <= 1e+225)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_6) * fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(2.0 * C))))) / Float64(-t_6));
	elseif (t_5 <= Inf)
		tmp = Float64(sqrt(Float64(F * t_0)) * Float64(t_1 * Float64(-1.0 / t_0)));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t$95$3 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$5, -1e-176], N[(N[Sqrt[N[(t$95$3 * N[(N[(N[(A + C), $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]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 1e+225], N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$6), $MachinePrecision] * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$6)), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $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

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

      1. lower-*.f64 17.7

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

   5. Applied rewrites 17.7%

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

   7. Applied rewrites 21.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

   9. Applied rewrites 28.9%

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

   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))) < -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. lower-/.f64 97.3

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

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 97.3

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 97.3

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

   4. Applied rewrites 97.3%

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

   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))) < 9.99999999999999928e224

   1. Initial program 23.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 40.5

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

   5. Applied rewrites 40.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

   7. Applied rewrites 40.5%

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

   if 9.99999999999999928e224 < (/.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 10.0%

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

   4. Applied rewrites 10.0%

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

   5. Taylor expanded in C around inf

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

      1. distribute-rgt-in N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 27.6

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

   7. Applied rewrites 27.6%

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

   9. Applied rewrites 45.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 18.0

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

   5. Applied rewrites 18.0%

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

   7. Applied rewrites 18.1%

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

   9. Applied rewrites 18.1%

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

   11. Applied rewrites 23.4%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(\sqrt{F} \cdot \sqrt{2 \cdot \left(2 \cdot C\right)}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-176}:\\ \;\;\;\;\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(A + C\right) \cdot \left(A - C\right), \frac{1}{A - C}, \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 10^{+225}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(\sqrt{2 \cdot \left(2 \cdot C\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \sqrt{2 \cdot \left(2 \cdot C\right)}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := t\_2 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ t_5 := \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{t\_0} \cdot \left(\sqrt{F} \cdot t\_1\right)}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{-t\_0}\\ \mathbf{elif}\;t\_4 \leq 10^{+225}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot t\_5\right) \cdot \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, 2 \cdot C\right)\right)}}{-t\_5}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{F \cdot t\_0} \cdot \left(t\_1 \cdot \frac{-1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (sqrt (* 2.0 (* 2.0 C))))
        (t_2 (* (* 4.0 A) C))
        (t_3 (- t_2 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_3))
        (t_5 (fma C (* A -4.0) (* B_m B_m))))
   (if (<= t_4 (- INFINITY))
     (/ (* (sqrt t_0) (* (sqrt F) t_1)) t_3)
     (if (<= t_4 -1e-176)
       (/
        (sqrt
         (*
          (* t_0 (* 2.0 F))
          (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (- t_0))
       (if (<= t_4 1e+225)
         (/
          (sqrt (* 2.0 (* (* F t_5) (fma (/ (* B_m B_m) A) -0.5 (* 2.0 C)))))
          (- t_5))
         (if (<= t_4 INFINITY)
           (* (sqrt (* F t_0)) (* t_1 (/ -1.0 t_0)))
           (- (/ (sqrt F) (sqrt (* B_m 0.5))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = sqrt((2.0 * (2.0 * C)));
	double t_2 = (4.0 * A) * C;
	double t_3 = t_2 - pow(B_m, 2.0);
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double t_5 = fma(C, (A * -4.0), (B_m * B_m));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt(t_0) * (sqrt(F) * t_1)) / t_3;
	} else if (t_4 <= -1e-176) {
		tmp = sqrt(((t_0 * (2.0 * F)) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / -t_0;
	} else if (t_4 <= 1e+225) {
		tmp = sqrt((2.0 * ((F * t_5) * fma(((B_m * B_m) / A), -0.5, (2.0 * C))))) / -t_5;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((F * t_0)) * (t_1 * (-1.0 / t_0));
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = sqrt(Float64(2.0 * Float64(2.0 * C)))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(t_2 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	t_5 = fma(C, Float64(A * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(t_0) * Float64(sqrt(F) * t_1)) / t_3);
	elseif (t_4 <= -1e-176)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / Float64(-t_0));
	elseif (t_4 <= 1e+225)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_5) * fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(2.0 * C))))) / Float64(-t_5));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(F * t_0)) * Float64(t_1 * Float64(-1.0 / t_0)));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -1e-176], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$4, 1e+225], N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$5), $MachinePrecision] * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$5)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $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

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

      1. lower-*.f64 17.7

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

   5. Applied rewrites 17.7%

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

   7. Applied rewrites 21.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

   9. Applied rewrites 28.9%

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

   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))) < -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

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

      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 97.2%

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

   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))) < 9.99999999999999928e224

   1. Initial program 23.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 40.5

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

   5. Applied rewrites 40.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

   7. Applied rewrites 40.5%

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

   if 9.99999999999999928e224 < (/.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 10.0%

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

   4. Applied rewrites 10.0%

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

   5. Taylor expanded in C around inf

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

      1. distribute-rgt-in N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 27.6

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

   7. Applied rewrites 27.6%

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

   9. Applied rewrites 45.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 18.0

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

   5. Applied rewrites 18.0%

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

   7. Applied rewrites 18.1%

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

   9. Applied rewrites 18.1%

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

   11. Applied rewrites 23.4%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(\sqrt{F} \cdot \sqrt{2 \cdot \left(2 \cdot C\right)}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 10^{+225}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(\sqrt{2 \cdot \left(2 \cdot C\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(\sqrt{t\_0} \cdot \left(\sqrt{2 \cdot F} \cdot \sqrt{2 \cdot C}\right)\right) \cdot \frac{-1}{t\_1}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq 10^{+225}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot t\_1\right) \cdot \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, 2 \cdot C\right)\right)}}{-t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{F \cdot t\_0} \cdot \left(\sqrt{2 \cdot \left(2 \cdot C\right)} \cdot \frac{-1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (fma C (* A -4.0) (* B_m B_m)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 (- INFINITY))
     (* (* (sqrt t_0) (* (sqrt (* 2.0 F)) (sqrt (* 2.0 C)))) (/ -1.0 t_1))
     (if (<= t_3 -1e-176)
       (/
        (sqrt
         (*
          (* t_0 (* 2.0 F))
          (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (- t_0))
       (if (<= t_3 1e+225)
         (/
          (sqrt (* 2.0 (* (* F t_1) (fma (/ (* B_m B_m) A) -0.5 (* 2.0 C)))))
          (- t_1))
         (if (<= t_3 INFINITY)
           (* (sqrt (* F t_0)) (* (sqrt (* 2.0 (* 2.0 C))) (/ -1.0 t_0)))
           (- (/ (sqrt F) (sqrt (* B_m 0.5))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = fma(C, (A * -4.0), (B_m * B_m));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt(t_0) * (sqrt((2.0 * F)) * sqrt((2.0 * C)))) * (-1.0 / t_1);
	} else if (t_3 <= -1e-176) {
		tmp = sqrt(((t_0 * (2.0 * F)) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / -t_0;
	} else if (t_3 <= 1e+225) {
		tmp = sqrt((2.0 * ((F * t_1) * fma(((B_m * B_m) / A), -0.5, (2.0 * C))))) / -t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((F * t_0)) * (sqrt((2.0 * (2.0 * C))) * (-1.0 / t_0));
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = fma(C, Float64(A * -4.0), Float64(B_m * B_m))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(t_0) * Float64(sqrt(Float64(2.0 * F)) * sqrt(Float64(2.0 * C)))) * Float64(-1.0 / t_1));
	elseif (t_3 <= -1e-176)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / Float64(-t_0));
	elseif (t_3 <= 1e+225)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_1) * fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(2.0 * C))))) / Float64(-t_1));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(F * t_0)) * Float64(sqrt(Float64(2.0 * Float64(2.0 * C))) * Float64(-1.0 / t_0)));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-176], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$3, 1e+225], N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $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

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

      1. lower-*.f64 17.7

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

   5. Applied rewrites 17.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 17.7%

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

   9. Applied rewrites 25.8%

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

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. unpow-prod-down N/A

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

       9. lower-*.f64 N/A

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

       10. pow1/2 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-*.f64 N/A

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

   11. Applied rewrites 29.0%

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

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

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

      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 97.2%

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

   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))) < 9.99999999999999928e224

   1. Initial program 23.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 40.5

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

   5. Applied rewrites 40.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

   7. Applied rewrites 40.5%

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

   if 9.99999999999999928e224 < (/.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 10.0%

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

   4. Applied rewrites 10.0%

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

   5. Taylor expanded in C around inf

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

      1. distribute-rgt-in N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 27.6

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

   7. Applied rewrites 27.6%

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

   9. Applied rewrites 45.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 18.0

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

   5. Applied rewrites 18.0%

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

   7. Applied rewrites 18.1%

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

   9. Applied rewrites 18.1%

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

   11. Applied rewrites 23.4%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(\sqrt{2 \cdot F} \cdot \sqrt{2 \cdot C}\right)\right) \cdot \frac{-1}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 10^{+225}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(\sqrt{2 \cdot \left(2 \cdot C\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\\ t_2 := \frac{-1}{t\_1}\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3 - {B\_m}^{2}}\\ t_5 := F \cdot t\_1\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\left(\sqrt{t\_0} \cdot \left(\sqrt{2 \cdot F} \cdot \sqrt{2 \cdot C}\right)\right) \cdot t\_2\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{2 \cdot \left(t\_5 \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B\_m \cdot B\_m\right)}\right)\right)} \cdot t\_2\\ \mathbf{elif}\;t\_4 \leq 10^{+225}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t\_5 \cdot \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, 2 \cdot C\right)\right)}}{-t\_1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{F \cdot t\_0} \cdot \left(\sqrt{2 \cdot \left(2 \cdot C\right)} \cdot \frac{-1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (fma C (* A -4.0) (* B_m B_m)))
        (t_2 (/ -1.0 t_1))
        (t_3 (* (* 4.0 A) C))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_3) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_3 (pow B_m 2.0))))
        (t_5 (* F t_1)))
   (if (<= t_4 (- INFINITY))
     (* (* (sqrt t_0) (* (sqrt (* 2.0 F)) (sqrt (* 2.0 C)))) t_2)
     (if (<= t_4 -1e-176)
       (* (sqrt (* 2.0 (* t_5 (+ C (sqrt (fma C C (* B_m B_m))))))) t_2)
       (if (<= t_4 1e+225)
         (/
          (sqrt (* 2.0 (* t_5 (fma (/ (* B_m B_m) A) -0.5 (* 2.0 C)))))
          (- t_1))
         (if (<= t_4 INFINITY)
           (* (sqrt (* F t_0)) (* (sqrt (* 2.0 (* 2.0 C))) (/ -1.0 t_0)))
           (- (/ (sqrt F) (sqrt (* B_m 0.5))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = fma(C, (A * -4.0), (B_m * B_m));
	double t_2 = -1.0 / t_1;
	double t_3 = (4.0 * A) * C;
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_3) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_3 - pow(B_m, 2.0));
	double t_5 = F * t_1;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt(t_0) * (sqrt((2.0 * F)) * sqrt((2.0 * C)))) * t_2;
	} else if (t_4 <= -1e-176) {
		tmp = sqrt((2.0 * (t_5 * (C + sqrt(fma(C, C, (B_m * B_m))))))) * t_2;
	} else if (t_4 <= 1e+225) {
		tmp = sqrt((2.0 * (t_5 * fma(((B_m * B_m) / A), -0.5, (2.0 * C))))) / -t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((F * t_0)) * (sqrt((2.0 * (2.0 * C))) * (-1.0 / t_0));
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = fma(C, Float64(A * -4.0), Float64(B_m * B_m))
	t_2 = Float64(-1.0 / t_1)
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_3) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_3 - (B_m ^ 2.0)))
	t_5 = Float64(F * t_1)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(t_0) * Float64(sqrt(Float64(2.0 * F)) * sqrt(Float64(2.0 * C)))) * t_2);
	elseif (t_4 <= -1e-176)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_5 * Float64(C + sqrt(fma(C, C, Float64(B_m * B_m))))))) * t_2);
	elseif (t_4 <= 1e+225)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_5 * fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(2.0 * C))))) / Float64(-t_1));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(F * t_0)) * Float64(sqrt(Float64(2.0 * Float64(2.0 * C))) * Float64(-1.0 / t_0)));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(F * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$4, -1e-176], N[(N[Sqrt[N[(2.0 * N[(t$95$5 * N[(C + N[Sqrt[N[(C * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 1e+225], N[(N[Sqrt[N[(2.0 * N[(t$95$5 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $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

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

      1. lower-*.f64 17.7

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

   5. Applied rewrites 17.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 17.7%

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

   9. Applied rewrites 25.8%

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

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. unpow-prod-down N/A

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

       9. lower-*.f64 N/A

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

       10. pow1/2 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-*.f64 N/A

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

   11. Applied rewrites 29.0%

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

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

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

      1. lower-*.f64 22.5

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

   5. Applied rewrites 22.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 22.5%

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

   8. Taylor expanded in A around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 71.0

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

   10. Applied rewrites 71.0%

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

   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))) < 9.99999999999999928e224

   1. Initial program 23.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 40.5

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

   5. Applied rewrites 40.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

   7. Applied rewrites 40.5%

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

   if 9.99999999999999928e224 < (/.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 10.0%

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

   4. Applied rewrites 10.0%

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

   5. Taylor expanded in C around inf

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

      1. distribute-rgt-in N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 27.6

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

   7. Applied rewrites 27.6%

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

   9. Applied rewrites 45.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 18.0

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

   5. Applied rewrites 18.0%

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

   7. Applied rewrites 18.1%

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

   9. Applied rewrites 18.1%

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

   11. Applied rewrites 23.4%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(\sqrt{2 \cdot F} \cdot \sqrt{2 \cdot C}\right)\right) \cdot \frac{-1}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 10^{+225}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(\sqrt{2 \cdot \left(2 \cdot C\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-106}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{t\_0 \cdot \left(F \cdot \left(2 \cdot C\right)\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{t\_0 \cdot \left(2 \cdot \left(2 \cdot C\right)\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= (pow B_m 2.0) 1e-106)
     (* (sqrt 2.0) (/ (sqrt (* t_0 (* F (* 2.0 C)))) (- t_0)))
     (if (<= (pow B_m 2.0) 2e+70)
       (*
        (* (sqrt F) (sqrt (* t_0 (* 2.0 (* 2.0 C)))))
        (/ -1.0 (fma C (* A -4.0) (* B_m B_m))))
       (- (/ (sqrt F) (sqrt (* B_m 0.5))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-106) {
		tmp = sqrt(2.0) * (sqrt((t_0 * (F * (2.0 * C)))) / -t_0);
	} else if (pow(B_m, 2.0) <= 2e+70) {
		tmp = (sqrt(F) * sqrt((t_0 * (2.0 * (2.0 * C))))) * (-1.0 / fma(C, (A * -4.0), (B_m * B_m)));
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-106)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(t_0 * Float64(F * Float64(2.0 * C)))) / Float64(-t_0)));
	elseif ((B_m ^ 2.0) <= 2e+70)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(t_0 * Float64(2.0 * Float64(2.0 * C))))) * Float64(-1.0 / fma(C, Float64(A * -4.0), Float64(B_m * B_m))));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-106], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(t$95$0 * N[(F * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+70], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(t$95$0 * N[(2.0 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999941e-107

   1. Initial program 21.4%

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

   4. Applied rewrites 22.0%

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

   5. Taylor expanded in C around inf

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

      1. distribute-rgt-in N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 30.7

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

   7. Applied rewrites 30.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 30.7

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

   9. Applied rewrites 30.7%

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

   if 9.99999999999999941e-107 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000015e70

   1. Initial program 31.0%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 19.3

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

   5. Applied rewrites 19.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 19.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-unprod N/A

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

   9. Applied rewrites 28.6%

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

   if 2.00000000000000015e70 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 13.6%

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

   3. Taylor expanded in 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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 26.0

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

   5. Applied rewrites 26.0%

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

   7. Applied rewrites 26.1%

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

   9. Applied rewrites 26.0%

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

   11. Applied rewrites 34.9%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-106}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot \left(2 \cdot C\right)\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-106}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\right)\right)} \cdot \left(-\frac{\frac{-0.25}{A}}{C}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B\_m \cdot B\_m\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-106)
   (*
    (sqrt (* 2.0 (* (* 2.0 C) (* F (fma C (* A -4.0) (* B_m B_m))))))
    (- (/ (/ -0.25 A) C)))
   (if (<= (pow B_m 2.0) 1e+99)
     (/ (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma C C (* B_m B_m))))))) (- B_m))
     (- (/ (sqrt F) (sqrt (* B_m 0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-106) {
		tmp = sqrt((2.0 * ((2.0 * C) * (F * fma(C, (A * -4.0), (B_m * B_m)))))) * -((-0.25 / A) / C);
	} else if (pow(B_m, 2.0) <= 1e+99) {
		tmp = (sqrt(2.0) * sqrt((F * (C + sqrt(fma(C, C, (B_m * B_m))))))) / -B_m;
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-106)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(2.0 * C) * Float64(F * fma(C, Float64(A * -4.0), Float64(B_m * B_m)))))) * Float64(-Float64(Float64(-0.25 / A) / C)));
	elseif ((B_m ^ 2.0) <= 1e+99)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + sqrt(fma(C, C, Float64(B_m * B_m))))))) / Float64(-B_m));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-106], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C), $MachinePrecision] * N[(F * N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[(-0.25 / A), $MachinePrecision] / C), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+99], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(C * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999941e-107

   1. Initial program 21.4%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 29.4

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

   5. Applied rewrites 29.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 29.2%

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

   8. Taylor expanded in C around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 28.3

         \[\leadsto \frac{-0.25}{\color{blue}{A \cdot C}} \cdot \left(-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   10. Applied rewrites 28.3%

       \[\leadsto \color{blue}{\frac{-0.25}{A \cdot C}} \cdot \left(-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 28.4%

       \[\leadsto \frac{\frac{-0.25}{A}}{\color{blue}{C}} \cdot \left(-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   if 9.99999999999999941e-107 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999997e98

   1. Initial program 31.5%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 18.7%

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

   if 9.9999999999999997e98 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.2%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 26.4

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

   5. Applied rewrites 26.4%

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

   7. Applied rewrites 26.5%

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

   9. Applied rewrites 26.5%

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

   11. Applied rewrites 36.0%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-106}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right)\right)} \cdot \left(-\frac{\frac{-0.25}{A}}{C}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{+99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-106}:\\ \;\;\;\;\sqrt{\left(2 \cdot C\right) \cdot \left(\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)} \cdot \left(-\frac{-0.25}{A \cdot C}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B\_m \cdot B\_m\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-106)
   (*
    (sqrt (* (* 2.0 C) (* (fma B_m B_m (* -4.0 (* A C))) (* 2.0 F))))
    (- (/ -0.25 (* A C))))
   (if (<= (pow B_m 2.0) 1e+99)
     (/ (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma C C (* B_m B_m))))))) (- B_m))
     (- (/ (sqrt F) (sqrt (* B_m 0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-106) {
		tmp = sqrt(((2.0 * C) * (fma(B_m, B_m, (-4.0 * (A * C))) * (2.0 * F)))) * -(-0.25 / (A * C));
	} else if (pow(B_m, 2.0) <= 1e+99) {
		tmp = (sqrt(2.0) * sqrt((F * (C + sqrt(fma(C, C, (B_m * B_m))))))) / -B_m;
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-106)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(fma(B_m, B_m, Float64(-4.0 * Float64(A * C))) * Float64(2.0 * F)))) * Float64(-Float64(-0.25 / Float64(A * C))));
	elseif ((B_m ^ 2.0) <= 1e+99)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + sqrt(fma(C, C, Float64(B_m * B_m))))))) / Float64(-B_m));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-106], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(-0.25 / N[(A * C), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+99], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(C * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999941e-107

   1. Initial program 21.4%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 29.4

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

   5. Applied rewrites 29.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 29.2%

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

   8. Taylor expanded in C around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 28.3

         \[\leadsto \frac{-0.25}{\color{blue}{A \cdot C}} \cdot \left(-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   10. Applied rewrites 28.3%

       \[\leadsto \color{blue}{\frac{-0.25}{A \cdot C}} \cdot \left(-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   12. Applied rewrites 28.3%

       \[\leadsto \color{blue}{\left(-\sqrt{\left(C \cdot 2\right) \cdot \left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot 2\right)\right)}\right) \cdot \frac{-0.25}{A \cdot C}} \]

   if 9.99999999999999941e-107 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999997e98

   1. Initial program 31.5%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 18.7%

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

   if 9.9999999999999997e98 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.2%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 26.4

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

   5. Applied rewrites 26.4%

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

   7. Applied rewrites 26.5%

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

   9. Applied rewrites 26.5%

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

   11. Applied rewrites 36.0%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-106}:\\ \;\;\;\;\sqrt{\left(2 \cdot C\right) \cdot \left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)} \cdot \left(-\frac{-0.25}{A \cdot C}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{+99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-106}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\right)\right)} \cdot \left(-\frac{-0.25}{A \cdot C}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B\_m \cdot B\_m\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-106)
   (*
    (sqrt (* 2.0 (* (* 2.0 C) (* F (fma C (* A -4.0) (* B_m B_m))))))
    (- (/ -0.25 (* A C))))
   (if (<= (pow B_m 2.0) 1e+99)
     (/ (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma C C (* B_m B_m))))))) (- B_m))
     (- (/ (sqrt F) (sqrt (* B_m 0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-106) {
		tmp = sqrt((2.0 * ((2.0 * C) * (F * fma(C, (A * -4.0), (B_m * B_m)))))) * -(-0.25 / (A * C));
	} else if (pow(B_m, 2.0) <= 1e+99) {
		tmp = (sqrt(2.0) * sqrt((F * (C + sqrt(fma(C, C, (B_m * B_m))))))) / -B_m;
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-106)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(2.0 * C) * Float64(F * fma(C, Float64(A * -4.0), Float64(B_m * B_m)))))) * Float64(-Float64(-0.25 / Float64(A * C))));
	elseif ((B_m ^ 2.0) <= 1e+99)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + sqrt(fma(C, C, Float64(B_m * B_m))))))) / Float64(-B_m));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-106], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C), $MachinePrecision] * N[(F * N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(-0.25 / N[(A * C), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+99], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(C * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999941e-107

   1. Initial program 21.4%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 29.4

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

   5. Applied rewrites 29.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 29.2%

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

   8. Taylor expanded in C around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 28.3

         \[\leadsto \frac{-0.25}{\color{blue}{A \cdot C}} \cdot \left(-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   10. Applied rewrites 28.3%

       \[\leadsto \color{blue}{\frac{-0.25}{A \cdot C}} \cdot \left(-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   if 9.99999999999999941e-107 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999997e98

   1. Initial program 31.5%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 18.7%

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

   if 9.9999999999999997e98 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.2%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 26.4

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

   5. Applied rewrites 26.4%

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

   7. Applied rewrites 26.5%

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

   9. Applied rewrites 26.5%

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

   11. Applied rewrites 36.0%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-106}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right)\right)} \cdot \left(-\frac{-0.25}{A \cdot C}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{+99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-106}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)} \cdot \left(-\frac{-0.25}{A \cdot C}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B\_m \cdot B\_m\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-106)
   (*
    (sqrt (* 2.0 (* (* 2.0 C) (* F (* -4.0 (* A C))))))
    (- (/ -0.25 (* A C))))
   (if (<= (pow B_m 2.0) 1e+99)
     (/ (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma C C (* B_m B_m))))))) (- B_m))
     (- (/ (sqrt F) (sqrt (* B_m 0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-106) {
		tmp = sqrt((2.0 * ((2.0 * C) * (F * (-4.0 * (A * C)))))) * -(-0.25 / (A * C));
	} else if (pow(B_m, 2.0) <= 1e+99) {
		tmp = (sqrt(2.0) * sqrt((F * (C + sqrt(fma(C, C, (B_m * B_m))))))) / -B_m;
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-106)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(2.0 * C) * Float64(F * Float64(-4.0 * Float64(A * C)))))) * Float64(-Float64(-0.25 / Float64(A * C))));
	elseif ((B_m ^ 2.0) <= 1e+99)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + sqrt(fma(C, C, Float64(B_m * B_m))))))) / Float64(-B_m));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-106], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C), $MachinePrecision] * N[(F * N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(-0.25 / N[(A * C), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+99], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(C * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999941e-107

   1. Initial program 21.4%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 29.4

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

   5. Applied rewrites 29.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 29.2%

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

   8. Taylor expanded in C around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 28.3

         \[\leadsto \frac{-0.25}{\color{blue}{A \cdot C}} \cdot \left(-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   10. Applied rewrites 28.3%

       \[\leadsto \color{blue}{\frac{-0.25}{A \cdot C}} \cdot \left(-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   11. Taylor expanded in C around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 28.3

          \[\leadsto \frac{-0.25}{A \cdot C} \cdot \left(-\sqrt{2 \cdot \left(\left(\left(-4 \cdot \color{blue}{\left(A \cdot C\right)}\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   13. Applied rewrites 28.3%

       \[\leadsto \frac{-0.25}{A \cdot C} \cdot \left(-\sqrt{2 \cdot \left(\left(\color{blue}{\left(-4 \cdot \left(A \cdot C\right)\right)} \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   if 9.99999999999999941e-107 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999997e98

   1. Initial program 31.5%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 18.7%

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

   if 9.9999999999999997e98 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.2%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 26.4

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

   5. Applied rewrites 26.4%

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

   7. Applied rewrites 26.5%

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

   9. Applied rewrites 26.5%

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

   11. Applied rewrites 36.0%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-106}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)} \cdot \left(-\frac{-0.25}{A \cdot C}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{+99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-106}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)} \cdot \left(-\frac{-0.25}{A \cdot C}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+99}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B\_m}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-106)
   (*
    (sqrt (* 2.0 (* (* 2.0 C) (* F (* -4.0 (* A C))))))
    (- (/ -0.25 (* A C))))
   (if (<= (pow B_m 2.0) 1e+99)
     (* (- (/ (sqrt 2.0) B_m)) (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C)))))))
     (- (/ (sqrt F) (sqrt (* B_m 0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-106) {
		tmp = sqrt((2.0 * ((2.0 * C) * (F * (-4.0 * (A * C)))))) * -(-0.25 / (A * C));
	} else if (pow(B_m, 2.0) <= 1e+99) {
		tmp = -(sqrt(2.0) / B_m) * sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C))))));
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-106)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(2.0 * C) * Float64(F * Float64(-4.0 * Float64(A * C)))))) * Float64(-Float64(-0.25 / Float64(A * C))));
	elseif ((B_m ^ 2.0) <= 1e+99)
		tmp = Float64(Float64(-Float64(sqrt(2.0) / B_m)) * sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-106], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C), $MachinePrecision] * N[(F * N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(-0.25 / N[(A * C), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+99], N[((-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]) * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999941e-107

   1. Initial program 21.4%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 29.4

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

   5. Applied rewrites 29.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 29.2%

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

   8. Taylor expanded in C around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 28.3

         \[\leadsto \frac{-0.25}{\color{blue}{A \cdot C}} \cdot \left(-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   10. Applied rewrites 28.3%

       \[\leadsto \color{blue}{\frac{-0.25}{A \cdot C}} \cdot \left(-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   11. Taylor expanded in C around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 28.3

          \[\leadsto \frac{-0.25}{A \cdot C} \cdot \left(-\sqrt{2 \cdot \left(\left(\left(-4 \cdot \color{blue}{\left(A \cdot C\right)}\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   13. Applied rewrites 28.3%

       \[\leadsto \frac{-0.25}{A \cdot C} \cdot \left(-\sqrt{2 \cdot \left(\left(\color{blue}{\left(-4 \cdot \left(A \cdot C\right)\right)} \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   if 9.99999999999999941e-107 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999997e98

   1. Initial program 31.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 15.2

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

   5. Applied rewrites 15.2%

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

   6. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 18.7

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

   8. Applied rewrites 18.7%

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

   if 9.9999999999999997e98 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.2%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 26.4

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

   5. Applied rewrites 26.4%

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

   7. Applied rewrites 26.5%

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

   9. Applied rewrites 26.5%

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

   11. Applied rewrites 36.0%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-106}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)} \cdot \left(-\frac{-0.25}{A \cdot C}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{+99}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{t\_0 \cdot \left(F \cdot \left(2 \cdot C\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= (pow B_m 2.0) 5e+33)
     (* (sqrt 2.0) (/ (sqrt (* t_0 (* F (* 2.0 C)))) (- t_0)))
     (- (/ (sqrt F) (sqrt (* B_m 0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+33) {
		tmp = sqrt(2.0) * (sqrt((t_0 * (F * (2.0 * C)))) / -t_0);
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+33)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(t_0 * Float64(F * Float64(2.0 * C)))) / Float64(-t_0)));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+33], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(t$95$0 * N[(F * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999973e33

   1. Initial program 22.9%

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in C around inf

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

      1. distribute-rgt-in N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 28.9

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

   7. Applied rewrites 28.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 28.9

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

   9. Applied rewrites 28.9%

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

   if 4.99999999999999973e33 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.8%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 25.3

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

   5. Applied rewrites 25.3%

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

   7. Applied rewrites 25.4%

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

   9. Applied rewrites 25.4%

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

   11. Applied rewrites 33.6%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= (pow B_m 2.0) 5e+33)
     (/ (sqrt (* (* 2.0 C) (* t_0 (* 2.0 F)))) (- t_0))
     (- (/ (sqrt F) (sqrt (* B_m 0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+33) {
		tmp = sqrt(((2.0 * C) * (t_0 * (2.0 * F)))) / -t_0;
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+33)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(t_0 * Float64(2.0 * F)))) / Float64(-t_0));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+33], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999973e33

   1. Initial program 22.9%

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in C around inf

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

      1. distribute-rgt-in N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 28.9

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

   7. Applied rewrites 28.9%

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

   9. Applied rewrites 27.3%

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

   if 4.99999999999999973e33 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.8%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 25.3

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

   5. Applied rewrites 25.3%

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

   7. Applied rewrites 25.4%

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

   9. Applied rewrites 25.4%

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

   11. Applied rewrites 33.6%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot t\_0\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma C (* A -4.0) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 5e+33)
     (/ (sqrt (* 2.0 (* (* 2.0 C) (* F t_0)))) (- t_0))
     (- (/ (sqrt F) (sqrt (* B_m 0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(C, (A * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+33) {
		tmp = sqrt((2.0 * ((2.0 * C) * (F * t_0)))) / -t_0;
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(C, Float64(A * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+33)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(2.0 * C) * Float64(F * t_0)))) / Float64(-t_0));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+33], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999973e33

   1. Initial program 22.9%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 27.3

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

   5. Applied rewrites 27.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 27.3%

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

   if 4.99999999999999973e33 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.8%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 25.3

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

   5. Applied rewrites 25.3%

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

   7. Applied rewrites 25.4%

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

   9. Applied rewrites 25.4%

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

   11. Applied rewrites 33.6%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)} \cdot \left(-\frac{-0.25}{A \cdot C}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e+21)
   (*
    (sqrt (* 2.0 (* (* 2.0 C) (* F (* -4.0 (* A C))))))
    (- (/ -0.25 (* A C))))
   (- (/ (sqrt F) (sqrt (* B_m 0.5))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e+21) {
		tmp = sqrt((2.0 * ((2.0 * C) * (F * (-4.0 * (A * C)))))) * -(-0.25 / (A * C));
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if ((b_m ** 2.0d0) <= 5d+21) then
        tmp = sqrt((2.0d0 * ((2.0d0 * c) * (f * ((-4.0d0) * (a * c)))))) * -((-0.25d0) / (a * c))
    else
        tmp = -(sqrt(f) / sqrt((b_m * 0.5d0)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e+21) {
		tmp = Math.sqrt((2.0 * ((2.0 * C) * (F * (-4.0 * (A * C)))))) * -(-0.25 / (A * C));
	} else {
		tmp = -(Math.sqrt(F) / Math.sqrt((B_m * 0.5)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e+21:
		tmp = math.sqrt((2.0 * ((2.0 * C) * (F * (-4.0 * (A * C)))))) * -(-0.25 / (A * C))
	else:
		tmp = -(math.sqrt(F) / math.sqrt((B_m * 0.5)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+21)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(2.0 * C) * Float64(F * Float64(-4.0 * Float64(A * C)))))) * Float64(-Float64(-0.25 / Float64(A * C))));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e+21)
		tmp = sqrt((2.0 * ((2.0 * C) * (F * (-4.0 * (A * C)))))) * -(-0.25 / (A * C));
	else
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+21], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C), $MachinePrecision] * N[(F * N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(-0.25 / N[(A * C), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e21

   1. Initial program 22.3%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 27.5

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

   5. Applied rewrites 27.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 27.4%

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

   8. Taylor expanded in C around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 25.9

         \[\leadsto \frac{-0.25}{\color{blue}{A \cdot C}} \cdot \left(-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   10. Applied rewrites 25.9%

       \[\leadsto \color{blue}{\frac{-0.25}{A \cdot C}} \cdot \left(-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   11. Taylor expanded in C around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 25.9

          \[\leadsto \frac{-0.25}{A \cdot C} \cdot \left(-\sqrt{2 \cdot \left(\left(\left(-4 \cdot \color{blue}{\left(A \cdot C\right)}\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   13. Applied rewrites 25.9%

       \[\leadsto \frac{-0.25}{A \cdot C} \cdot \left(-\sqrt{2 \cdot \left(\left(\color{blue}{\left(-4 \cdot \left(A \cdot C\right)\right)} \cdot F\right) \cdot \left(2 \cdot C\right)\right)}\right) \]

   if 5e21 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 16.9%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 25.2

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

   5. Applied rewrites 25.2%

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

   7. Applied rewrites 25.3%

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

   9. Applied rewrites 25.3%

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

   11. Applied rewrites 33.2%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)} \cdot \left(-\frac{-0.25}{A \cdot C}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (/ (sqrt F) (sqrt (* B_m 0.5)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -(sqrt(F) / sqrt((B_m * 0.5)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -(sqrt(f) / sqrt((b_m * 0.5d0)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -(Math.sqrt(F) / Math.sqrt((B_m * 0.5)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -(math.sqrt(F) / math.sqrt((B_m * 0.5)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 20.0%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 13.8

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

5. Applied rewrites 13.8%

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

7. Applied rewrites 13.8%

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

9. Applied rewrites 13.8%

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

11. Applied rewrites 17.3%

    \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
12. Add Preprocessing

Alternative 16: 35.2% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((2.0d0 / b_m)) * -sqrt(f)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 13.8

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

5. Applied rewrites 13.8%

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

7. Applied rewrites 13.8%

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

9. Applied rewrites 17.2%

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

10. Final simplification 17.2%

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

Alternative 17: 27.2% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(((2.0d0 * f) / b_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 13.8

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

5. Applied rewrites 13.8%

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

7. Applied rewrites 13.8%

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

Alternative 18: 27.1% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt((f * (2.0d0 / b_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 13.8

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

5. Applied rewrites 13.8%

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

7. Applied rewrites 13.8%

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

9. Applied rewrites 13.8%

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