ABCF->ab-angle a

Percentage Accurate: 19.7% → 57.2%

Time: 22.8s

Alternatives: 14

Speedup: 6.0×

• Could not converge on a ground truth

  1. A = -2.2187490150535405e-296
  2. B = 3.039634558207673e-187
  3. C = 7.017380847215894e-130
  4. F = 25007.605179546106

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 57.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{{B\_m}^{2}}{A}\\ t_2 := 4 \cdot C - t\_1\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3 - {B\_m}^{2}}\\ t_5 := F \cdot t\_0\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{+239}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{t\_2}{\mathsf{fma}\left(A, C \cdot -4, {B\_m}^{2}\right)}}\right)\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{\left(2 \cdot t\_5\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{-1}{A \cdot \frac{\mathsf{fma}\left(C, -4, t\_1\right)}{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)\right) \cdot t\_2}}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_5} \cdot \sqrt{4 \cdot C}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (/ (pow B_m 2.0) A))
        (t_2 (- (* 4.0 C) 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_0)))
   (if (<= t_4 -1e+239)
     (* (sqrt F) (- (sqrt (/ t_2 (fma A (* C -4.0) (pow B_m 2.0))))))
     (if (<= t_4 -1e-191)
       (* (sqrt (* (* 2.0 t_5) (+ (+ A C) (hypot (- A C) B_m)))) (/ -1.0 t_0))
       (if (<= t_4 5e-132)
         (/
          -1.0
          (*
           A
           (/
            (fma C -4.0 t_1)
            (sqrt (* (* F (fma -4.0 (* A C) (pow B_m 2.0))) t_2)))))
         (if (<= t_4 INFINITY)
           (/ (* (sqrt t_5) (sqrt (* 4.0 C))) (- t_0))
           (* (sqrt F) (- (sqrt (/ 2.0 B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = pow(B_m, 2.0) / A;
	double t_2 = (4.0 * C) - 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_0;
	double tmp;
	if (t_4 <= -1e+239) {
		tmp = sqrt(F) * -sqrt((t_2 / fma(A, (C * -4.0), pow(B_m, 2.0))));
	} else if (t_4 <= -1e-191) {
		tmp = sqrt(((2.0 * t_5) * ((A + C) + hypot((A - C), B_m)))) * (-1.0 / t_0);
	} else if (t_4 <= 5e-132) {
		tmp = -1.0 / (A * (fma(C, -4.0, t_1) / sqrt(((F * fma(-4.0, (A * C), pow(B_m, 2.0))) * t_2))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt(t_5) * sqrt((4.0 * C))) / -t_0;
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64((B_m ^ 2.0) / A)
	t_2 = Float64(Float64(4.0 * C) - 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_0)
	tmp = 0.0
	if (t_4 <= -1e+239)
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(t_2 / fma(A, Float64(C * -4.0), (B_m ^ 2.0))))));
	elseif (t_4 <= -1e-191)
		tmp = Float64(sqrt(Float64(Float64(2.0 * t_5) * Float64(Float64(A + C) + hypot(Float64(A - C), B_m)))) * Float64(-1.0 / t_0));
	elseif (t_4 <= 5e-132)
		tmp = Float64(-1.0 / Float64(A * Float64(fma(C, -4.0, t_1) / sqrt(Float64(Float64(F * fma(-4.0, Float64(A * C), (B_m ^ 2.0))) * t_2)))));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(t_5) * sqrt(Float64(4.0 * C))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * C), $MachinePrecision] - 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$0), $MachinePrecision]}, If[LessEqual[t$95$4, -1e+239], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(t$95$2 / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$4, -1e-191], N[(N[Sqrt[N[(N[(2.0 * t$95$5), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e-132], N[(-1.0 / N[(A * N[(N[(C * -4.0 + t$95$1), $MachinePrecision] / N[Sqrt[N[(N[(F * N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[t$95$5], $MachinePrecision] * N[Sqrt[N[(4.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 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))) < -9.99999999999999991e238

   1. Initial program 5.6%

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

   3. Simplified 17.8%

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

   5. Taylor expanded in A around -inf 34.3%

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

   6. Taylor expanded in F around 0 25.5%

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

      1. pow1/2 25.8%

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

      2. associate-/l* 31.1%

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

      3. unpow-prod-down 49.5%

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

      4. pow1/2 49.5%

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

      5. +-commutative 49.5%

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

      6. mul-1-neg 49.5%

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

      7. sub-neg 49.5%

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

      8. *-commutative 49.5%

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

      9. *-commutative 49.5%

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

      10. associate-*r* 49.5%

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

      11. fma-define 49.5%

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

   8. Applied egg-rr 49.5%

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{C \cdot 4 - \frac{{B}^{2}}{A}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\right)}^{0.5}\right)} \]
   9. Step-by-step derivation

      1. unpow1/2 49.3%

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

   10. Simplified 49.3%

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

   if -9.99999999999999991e238 < (/.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-191

   1. Initial program 97.2%

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

   3. Simplified 97.3%

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

      1. associate-*r* 97.3%

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

      2. associate-+r+ 97.3%

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

      3. hypot-undefine 97.2%

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

      4. unpow2 97.2%

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

      5. unpow2 97.2%

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

      6. +-commutative 97.2%

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

      7. sqrt-prod 97.3%

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

      8. *-commutative 97.3%

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

      9. associate-+l+ 97.3%

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

   6. Applied egg-rr 97.3%

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

      1. div-inv 97.3%

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

      2. sqrt-unprod 97.4%

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

      3. *-commutative 97.4%

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

      4. associate-+r+ 97.4%

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

   8. Applied egg-rr 97.4%

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

   if -1e-191 < (/.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))) < 4.9999999999999999e-132

   1. Initial program 24.7%

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

   3. Simplified 24.5%

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

   5. Taylor expanded in A around -inf 28.9%

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

   6. Taylor expanded in A around -inf 28.9%

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

      1. associate-*r* 28.9%

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

      2. mul-1-neg 28.9%

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

      3. cancel-sign-sub-inv 28.9%

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

      4. metadata-eval 28.9%

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

   8. Simplified 28.9%

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

      1. clear-num 29.0%

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

      2. inv-pow 29.0%

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

   10. Applied egg-rr 29.0%

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

       1. unpow-1 29.0%

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

       2. associate-/l* 29.0%

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

       3. fma-undefine 29.0%

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

       4. unpow2 29.0%

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

       5. associate-*r* 29.0%

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

       6. *-commutative 29.0%

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

       7. +-commutative 29.0%

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

       8. fma-define 29.0%

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

       9. *-commutative 29.0%

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

       10. distribute-frac-neg 29.0%

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

       11. fmm-undef 29.0%

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

   12. Simplified 29.0%

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

   if 4.9999999999999999e-132 < (/.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 27.6%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in A around -inf 40.0%

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

      1. *-commutative 40.0%

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

   7. Simplified 40.0%

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

      1. sqrt-prod 56.2%

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

      2. *-commutative 56.2%

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

      3. *-commutative 56.2%

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

   9. Applied egg-rr 56.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf 20.3%

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

      1. mul-1-neg 20.3%

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

      2. *-commutative 20.3%

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

   5. Simplified 20.3%

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

      1. *-commutative 20.3%

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

      2. pow1/2 20.4%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 20.4%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 20.6%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 20.6%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 20.5%

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

   9. Simplified 20.5%

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

       1. associate-*l/ 20.5%

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

   11. Applied egg-rr 20.5%

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

       1. pow1/2 20.6%

          \[\leadsto -\color{blue}{{\left(\frac{F \cdot 2}{B}\right)}^{0.5}} \]

       2. associate-/l* 20.6%

          \[\leadsto -{\color{blue}{\left(F \cdot \frac{2}{B}\right)}}^{0.5} \]

       3. unpow-prod-down 28.5%

          \[\leadsto -\color{blue}{{F}^{0.5} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]

       4. pow1/2 28.5%

          \[\leadsto -\color{blue}{\sqrt{F}} \cdot {\left(\frac{2}{B}\right)}^{0.5} \]

   13. Applied egg-rr 28.5%

       \[\leadsto -\color{blue}{\sqrt{F} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]
   14. Step-by-step derivation

       1. unpow1/2 28.5%

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

   15. Simplified 28.5%

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

4. Final simplification 42.2%

   \[\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 -1 \cdot 10^{+239}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{4 \cdot C - \frac{{B}^{2}}{A}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\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 -1 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\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 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{-1}{A \cdot \frac{\mathsf{fma}\left(C, -4, \frac{{B}^{2}}{A}\right)}{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)\right) \cdot \left(4 \cdot C - \frac{{B}^{2}}{A}\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{4 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{{B\_m}^{2}}{A}\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-1}{A \cdot \frac{\mathsf{fma}\left(C, -4, t\_0\right)}{\sqrt{\left(4 \cdot C - t\_0\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t\_1\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{-t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\frac{-1}{A \cdot \left(\sqrt{\frac{-1}{F \cdot \left(\left({B\_m}^{2} + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(t\_0 - 4 \cdot C\right)\right)}} \cdot \left(t\_0 + C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (pow B_m 2.0) A)) (t_1 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 5e-176)
     (/
      -1.0
      (*
       A
       (/
        (fma C -4.0 t_0)
        (sqrt (* (- (* 4.0 C) t_0) (* -4.0 (* A (* C F))))))))
     (if (<= (pow B_m 2.0) 2e+42)
       (/
        (* (sqrt (* 2.0 (* F t_1))) (sqrt (+ A (+ C (hypot (- A C) B_m)))))
        (- t_1))
       (if (<= (pow B_m 2.0) 5e+111)
         (/
          -1.0
          (*
           A
           (*
            (sqrt
             (/
              -1.0
              (* F (* (+ (pow B_m 2.0) (* -4.0 (* A C))) (- t_0 (* 4.0 C))))))
            (+ t_0 (* C -4.0)))))
         (if (<= (pow B_m 2.0) 4e+259)
           (*
            (sqrt 2.0)
            (-
             (sqrt
              (*
               F
               (/
                (+ (+ A C) (hypot B_m (- A C)))
                (fma -4.0 (* A C) (pow B_m 2.0)))))))
           (* (sqrt F) (- (sqrt (/ 2.0 B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) / A;
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-176) {
		tmp = -1.0 / (A * (fma(C, -4.0, t_0) / sqrt((((4.0 * C) - t_0) * (-4.0 * (A * (C * F)))))));
	} else if (pow(B_m, 2.0) <= 2e+42) {
		tmp = (sqrt((2.0 * (F * t_1))) * sqrt((A + (C + hypot((A - C), B_m))))) / -t_1;
	} else if (pow(B_m, 2.0) <= 5e+111) {
		tmp = -1.0 / (A * (sqrt((-1.0 / (F * ((pow(B_m, 2.0) + (-4.0 * (A * C))) * (t_0 - (4.0 * C)))))) * (t_0 + (C * -4.0))));
	} else if (pow(B_m, 2.0) <= 4e+259) {
		tmp = sqrt(2.0) * -sqrt((F * (((A + C) + hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0)))));
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) / A)
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-176)
		tmp = Float64(-1.0 / Float64(A * Float64(fma(C, -4.0, t_0) / sqrt(Float64(Float64(Float64(4.0 * C) - t_0) * Float64(-4.0 * Float64(A * Float64(C * F))))))));
	elseif ((B_m ^ 2.0) <= 2e+42)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_1))) * sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m))))) / Float64(-t_1));
	elseif ((B_m ^ 2.0) <= 5e+111)
		tmp = Float64(-1.0 / Float64(A * Float64(sqrt(Float64(-1.0 / Float64(F * Float64(Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C))) * Float64(t_0 - Float64(4.0 * C)))))) * Float64(t_0 + Float64(C * -4.0)))));
	elseif ((B_m ^ 2.0) <= 4e+259)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))))));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-176], N[(-1.0 / N[(A * N[(N[(C * -4.0 + t$95$0), $MachinePrecision] / N[Sqrt[N[(N[(N[(4.0 * C), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+42], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+111], N[(-1.0 / N[(A * N[(N[Sqrt[N[(-1.0 / N[(F * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 + N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+259], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e-176

   1. Initial program 19.7%

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

   3. Simplified 31.3%

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

   5. Taylor expanded in A around -inf 22.2%

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

   6. Taylor expanded in A around -inf 22.2%

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

      1. associate-*r* 22.2%

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

      2. mul-1-neg 22.2%

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

      3. cancel-sign-sub-inv 22.2%

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

      4. metadata-eval 22.2%

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

   8. Simplified 22.2%

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

      1. clear-num 22.2%

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

      2. inv-pow 22.2%

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

   10. Applied egg-rr 22.2%

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

       1. unpow-1 22.2%

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

       2. associate-/l* 22.6%

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

       3. fma-undefine 22.6%

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

       4. unpow2 22.6%

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

       5. associate-*r* 22.6%

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

       6. *-commutative 22.6%

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

       7. +-commutative 22.6%

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

       8. fma-define 22.6%

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

       9. *-commutative 22.6%

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

       10. distribute-frac-neg 22.6%

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

       11. fmm-undef 22.6%

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

   12. Simplified 22.6%

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

   13. Taylor expanded in C around inf 22.0%

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

   if 5e-176 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000009e42

   1. Initial program 45.3%

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

   3. Simplified 50.4%

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

      1. associate-*r* 50.4%

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

      2. associate-+r+ 50.0%

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

      3. hypot-undefine 45.3%

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

      4. unpow2 45.3%

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

      5. unpow2 45.3%

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

      6. +-commutative 45.3%

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

      7. sqrt-prod 51.7%

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

      8. *-commutative 51.7%

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

      9. associate-+l+ 52.1%

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

   6. Applied egg-rr 67.9%

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

   if 2.00000000000000009e42 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999997e111

   1. Initial program 8.4%

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

   3. Simplified 10.2%

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

   5. Taylor expanded in A around -inf 34.1%

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

   6. Taylor expanded in A around -inf 33.9%

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

      1. associate-*r* 33.9%

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

      2. mul-1-neg 33.9%

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

      3. cancel-sign-sub-inv 33.9%

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

      4. metadata-eval 33.9%

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

   8. Simplified 33.9%

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

      1. clear-num 33.8%

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

      2. inv-pow 33.8%

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

   10. Applied egg-rr 33.8%

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

       1. unpow-1 33.8%

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

       2. associate-/l* 33.9%

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

       3. fma-undefine 33.9%

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

       4. unpow2 33.9%

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

       5. associate-*r* 33.9%

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

       6. *-commutative 33.9%

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

       7. +-commutative 33.9%

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

       8. fma-define 33.9%

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

       9. *-commutative 33.9%

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

       10. distribute-frac-neg 33.9%

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

       11. fmm-undef 33.9%

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

   12. Simplified 33.9%

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

   13. Taylor expanded in F around 0 40.4%

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

   if 4.9999999999999997e111 < (pow.f64 B #s(literal 2 binary64)) < 4e259

   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 F around 0 27.4%

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

   5. Simplified 57.9%

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

   if 4e259 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 2.9%

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

   3. Taylor expanded in B around inf 32.2%

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

      1. mul-1-neg 32.2%

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

      2. *-commutative 32.2%

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

   5. Simplified 32.2%

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

      1. *-commutative 32.2%

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

      2. pow1/2 32.2%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 32.2%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 32.4%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 32.4%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 32.4%

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

   9. Simplified 32.4%

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

       1. associate-*l/ 32.4%

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

   11. Applied egg-rr 32.4%

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

       1. pow1/2 32.4%

          \[\leadsto -\color{blue}{{\left(\frac{F \cdot 2}{B}\right)}^{0.5}} \]

       2. associate-/l* 32.4%

          \[\leadsto -{\color{blue}{\left(F \cdot \frac{2}{B}\right)}}^{0.5} \]

       3. unpow-prod-down 45.6%

          \[\leadsto -\color{blue}{{F}^{0.5} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]

       4. pow1/2 45.6%

          \[\leadsto -\color{blue}{\sqrt{F}} \cdot {\left(\frac{2}{B}\right)}^{0.5} \]

   13. Applied egg-rr 45.6%

       \[\leadsto -\color{blue}{\sqrt{F} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]
   14. Step-by-step derivation

       1. unpow1/2 45.6%

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

   15. Simplified 45.6%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-1}{A \cdot \frac{\mathsf{fma}\left(C, -4, \frac{{B}^{2}}{A}\right)}{\sqrt{\left(4 \cdot C - \frac{{B}^{2}}{A}\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\frac{-1}{A \cdot \left(\sqrt{\frac{-1}{F \cdot \left(\left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\frac{{B}^{2}}{A} - 4 \cdot C\right)\right)}} \cdot \left(\frac{{B}^{2}}{A} + C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{{B\_m}^{2}}{A}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-67}:\\ \;\;\;\;\frac{-1}{A \cdot \frac{\mathsf{fma}\left(C, -4, t\_0\right)}{\sqrt{\left(4 \cdot C - t\_0\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (pow B_m 2.0) A)))
   (if (<= (pow B_m 2.0) 1e-67)
     (/
      -1.0
      (*
       A
       (/
        (fma C -4.0 t_0)
        (sqrt (* (- (* 4.0 C) t_0) (* -4.0 (* A (* C F))))))))
     (if (<= (pow B_m 2.0) 4e+259)
       (*
        (sqrt 2.0)
        (-
         (sqrt
          (*
           F
           (/
            (+ (+ A C) (hypot B_m (- A C)))
            (fma -4.0 (* A C) (pow B_m 2.0)))))))
       (* (sqrt F) (- (sqrt (/ 2.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) / A;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-67) {
		tmp = -1.0 / (A * (fma(C, -4.0, t_0) / sqrt((((4.0 * C) - t_0) * (-4.0 * (A * (C * F)))))));
	} else if (pow(B_m, 2.0) <= 4e+259) {
		tmp = sqrt(2.0) * -sqrt((F * (((A + C) + hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0)))));
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) / A)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-67)
		tmp = Float64(-1.0 / Float64(A * Float64(fma(C, -4.0, t_0) / sqrt(Float64(Float64(Float64(4.0 * C) - t_0) * Float64(-4.0 * Float64(A * Float64(C * F))))))));
	elseif ((B_m ^ 2.0) <= 4e+259)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))))));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-67], N[(-1.0 / N[(A * N[(N[(C * -4.0 + t$95$0), $MachinePrecision] / N[Sqrt[N[(N[(N[(4.0 * C), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+259], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999943e-68

   1. Initial program 22.4%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in A around -inf 24.5%

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

   6. Taylor expanded in A around -inf 24.5%

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

      1. associate-*r* 24.5%

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

      2. mul-1-neg 24.5%

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

      3. cancel-sign-sub-inv 24.5%

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

      4. metadata-eval 24.5%

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

   8. Simplified 24.5%

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

      1. clear-num 24.5%

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

      2. inv-pow 24.5%

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

   10. Applied egg-rr 24.5%

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

       1. unpow-1 24.5%

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

       2. associate-/l* 24.9%

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

       3. fma-undefine 24.9%

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

       4. unpow2 24.9%

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

       5. associate-*r* 24.9%

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

       6. *-commutative 24.9%

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

       7. +-commutative 24.9%

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

       8. fma-define 24.9%

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

       9. *-commutative 24.9%

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

       10. distribute-frac-neg 24.9%

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

       11. fmm-undef 24.9%

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

   12. Simplified 24.9%

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

   13. Taylor expanded in C around inf 25.2%

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

   if 9.99999999999999943e-68 < (pow.f64 B #s(literal 2 binary64)) < 4e259

   1. Initial program 30.6%

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

   3. Taylor expanded in F around 0 32.2%

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

   5. Simplified 48.6%

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

   if 4e259 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 2.9%

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

   3. Taylor expanded in B around inf 32.2%

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

      1. mul-1-neg 32.2%

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

      2. *-commutative 32.2%

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

   5. Simplified 32.2%

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

      1. *-commutative 32.2%

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

      2. pow1/2 32.2%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 32.2%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 32.4%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 32.4%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 32.4%

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

   9. Simplified 32.4%

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

       1. associate-*l/ 32.4%

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

   11. Applied egg-rr 32.4%

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

       1. pow1/2 32.4%

          \[\leadsto -\color{blue}{{\left(\frac{F \cdot 2}{B}\right)}^{0.5}} \]

       2. associate-/l* 32.4%

          \[\leadsto -{\color{blue}{\left(F \cdot \frac{2}{B}\right)}}^{0.5} \]

       3. unpow-prod-down 45.6%

          \[\leadsto -\color{blue}{{F}^{0.5} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]

       4. pow1/2 45.6%

          \[\leadsto -\color{blue}{\sqrt{F}} \cdot {\left(\frac{2}{B}\right)}^{0.5} \]

   13. Applied egg-rr 45.6%

       \[\leadsto -\color{blue}{\sqrt{F} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]
   14. Step-by-step derivation

       1. unpow1/2 45.6%

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

   15. Simplified 45.6%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-67}:\\ \;\;\;\;\frac{-1}{A \cdot \frac{\mathsf{fma}\left(C, -4, \frac{{B}^{2}}{A}\right)}{\sqrt{\left(4 \cdot C - \frac{{B}^{2}}{A}\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{{B\_m}^{2}}{A}\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-1}{A \cdot \frac{\mathsf{fma}\left(C, -4, t\_0\right)}{\sqrt{\left(4 \cdot C - t\_0\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t\_1\right)} \cdot \sqrt{2 \cdot C}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (pow B_m 2.0) A)) (t_1 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 5e-176)
     (/
      -1.0
      (*
       A
       (/
        (fma C -4.0 t_0)
        (sqrt (* (- (* 4.0 C) t_0) (* -4.0 (* A (* C F))))))))
     (if (<= (pow B_m 2.0) 4e+221)
       (/ (* (sqrt (* 2.0 (* F t_1))) (sqrt (* 2.0 C))) (- t_1))
       (* (sqrt F) (- (sqrt (/ 2.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) / A;
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-176) {
		tmp = -1.0 / (A * (fma(C, -4.0, t_0) / sqrt((((4.0 * C) - t_0) * (-4.0 * (A * (C * F)))))));
	} else if (pow(B_m, 2.0) <= 4e+221) {
		tmp = (sqrt((2.0 * (F * t_1))) * sqrt((2.0 * C))) / -t_1;
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) / A)
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-176)
		tmp = Float64(-1.0 / Float64(A * Float64(fma(C, -4.0, t_0) / sqrt(Float64(Float64(Float64(4.0 * C) - t_0) * Float64(-4.0 * Float64(A * Float64(C * F))))))));
	elseif ((B_m ^ 2.0) <= 4e+221)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_1))) * sqrt(Float64(2.0 * C))) / Float64(-t_1));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-176], N[(-1.0 / N[(A * N[(N[(C * -4.0 + t$95$0), $MachinePrecision] / N[Sqrt[N[(N[(N[(4.0 * C), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+221], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e-176

   1. Initial program 19.7%

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

   3. Simplified 31.3%

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

   5. Taylor expanded in A around -inf 22.2%

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

   6. Taylor expanded in A around -inf 22.2%

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

      1. associate-*r* 22.2%

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

      2. mul-1-neg 22.2%

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

      3. cancel-sign-sub-inv 22.2%

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

      4. metadata-eval 22.2%

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

   8. Simplified 22.2%

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

      1. clear-num 22.2%

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

      2. inv-pow 22.2%

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

   10. Applied egg-rr 22.2%

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

       1. unpow-1 22.2%

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

       2. associate-/l* 22.6%

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

       3. fma-undefine 22.6%

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

       4. unpow2 22.6%

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

       5. associate-*r* 22.6%

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

       6. *-commutative 22.6%

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

       7. +-commutative 22.6%

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

       8. fma-define 22.6%

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

       9. *-commutative 22.6%

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

       10. distribute-frac-neg 22.6%

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

       11. fmm-undef 22.6%

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

   12. Simplified 22.6%

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

   13. Taylor expanded in C around inf 22.0%

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

   if 5e-176 < (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000002e221

   1. Initial program 34.5%

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

   3. Simplified 40.7%

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

      1. associate-*r* 40.7%

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

      2. associate-+r+ 40.0%

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

      3. hypot-undefine 34.5%

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

      4. unpow2 34.5%

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

      5. unpow2 34.5%

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

      6. +-commutative 34.5%

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

      7. sqrt-prod 38.1%

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

      8. *-commutative 38.1%

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

      9. associate-+l+ 38.6%

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

   6. Applied egg-rr 52.8%

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

   7. Taylor expanded in A around -inf 28.5%

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

   if 4.0000000000000002e221 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 2.9%

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

   3. Taylor expanded in B around inf 31.6%

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

      1. mul-1-neg 31.6%

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

      2. *-commutative 31.6%

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

   5. Simplified 31.6%

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

      1. *-commutative 31.6%

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

      2. pow1/2 31.6%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 31.6%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 31.9%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 31.9%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 31.9%

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

   9. Simplified 31.9%

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

       1. associate-*l/ 31.9%

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

   11. Applied egg-rr 31.9%

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

       1. pow1/2 31.9%

          \[\leadsto -\color{blue}{{\left(\frac{F \cdot 2}{B}\right)}^{0.5}} \]

       2. associate-/l* 31.9%

          \[\leadsto -{\color{blue}{\left(F \cdot \frac{2}{B}\right)}}^{0.5} \]

       3. unpow-prod-down 44.0%

          \[\leadsto -\color{blue}{{F}^{0.5} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]

       4. pow1/2 44.0%

          \[\leadsto -\color{blue}{\sqrt{F}} \cdot {\left(\frac{2}{B}\right)}^{0.5} \]

   13. Applied egg-rr 44.0%

       \[\leadsto -\color{blue}{\sqrt{F} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]
   14. Step-by-step derivation

       1. unpow1/2 44.0%

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

   15. Simplified 44.0%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-1}{A \cdot \frac{\mathsf{fma}\left(C, -4, \frac{{B}^{2}}{A}\right)}{\sqrt{\left(4 \cdot C - \frac{{B}^{2}}{A}\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-1}{A \cdot \frac{C \cdot -4}{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)\right) \cdot \left(4 \cdot C - \frac{{B\_m}^{2}}{A}\right)}}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t\_0\right)} \cdot \sqrt{2 \cdot C}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 5e-176)
     (/
      -1.0
      (*
       A
       (/
        (* C -4.0)
        (sqrt
         (*
          (* F (fma -4.0 (* A C) (pow B_m 2.0)))
          (- (* 4.0 C) (/ (pow B_m 2.0) A)))))))
     (if (<= (pow B_m 2.0) 4e+221)
       (/ (* (sqrt (* 2.0 (* F t_0))) (sqrt (* 2.0 C))) (- t_0))
       (* (sqrt F) (- (sqrt (/ 2.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-176) {
		tmp = -1.0 / (A * ((C * -4.0) / sqrt(((F * fma(-4.0, (A * C), pow(B_m, 2.0))) * ((4.0 * C) - (pow(B_m, 2.0) / A))))));
	} else if (pow(B_m, 2.0) <= 4e+221) {
		tmp = (sqrt((2.0 * (F * t_0))) * sqrt((2.0 * C))) / -t_0;
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-176)
		tmp = Float64(-1.0 / Float64(A * Float64(Float64(C * -4.0) / sqrt(Float64(Float64(F * fma(-4.0, Float64(A * C), (B_m ^ 2.0))) * Float64(Float64(4.0 * C) - Float64((B_m ^ 2.0) / A)))))));
	elseif ((B_m ^ 2.0) <= 4e+221)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * sqrt(Float64(2.0 * C))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-176], N[(-1.0 / N[(A * N[(N[(C * -4.0), $MachinePrecision] / N[Sqrt[N[(N[(F * N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(4.0 * C), $MachinePrecision] - N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+221], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e-176

   1. Initial program 19.7%

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

   3. Simplified 31.3%

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

   5. Taylor expanded in A around -inf 22.2%

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

   6. Taylor expanded in A around -inf 22.2%

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

      1. associate-*r* 22.2%

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

      2. mul-1-neg 22.2%

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

      3. cancel-sign-sub-inv 22.2%

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

      4. metadata-eval 22.2%

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

   8. Simplified 22.2%

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

      1. clear-num 22.2%

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

      2. inv-pow 22.2%

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

   10. Applied egg-rr 22.2%

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

       1. unpow-1 22.2%

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

       2. associate-/l* 22.6%

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

       3. fma-undefine 22.6%

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

       4. unpow2 22.6%

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

       5. associate-*r* 22.6%

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

       6. *-commutative 22.6%

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

       7. +-commutative 22.6%

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

       8. fma-define 22.6%

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

       9. *-commutative 22.6%

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

       10. distribute-frac-neg 22.6%

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

       11. fmm-undef 22.6%

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

   12. Simplified 22.6%

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

   13. Taylor expanded in C around inf 22.5%

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

   if 5e-176 < (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000002e221

   1. Initial program 34.5%

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

   3. Simplified 40.7%

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

      1. associate-*r* 40.7%

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

      2. associate-+r+ 40.0%

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

      3. hypot-undefine 34.5%

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

      4. unpow2 34.5%

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

      5. unpow2 34.5%

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

      6. +-commutative 34.5%

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

      7. sqrt-prod 38.1%

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

      8. *-commutative 38.1%

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

      9. associate-+l+ 38.6%

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

   6. Applied egg-rr 52.8%

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

   7. Taylor expanded in A around -inf 28.5%

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

   if 4.0000000000000002e221 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 2.9%

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

   3. Taylor expanded in B around inf 31.6%

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

      1. mul-1-neg 31.6%

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

      2. *-commutative 31.6%

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

   5. Simplified 31.6%

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

      1. *-commutative 31.6%

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

      2. pow1/2 31.6%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 31.6%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 31.9%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 31.9%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 31.9%

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

   9. Simplified 31.9%

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

       1. associate-*l/ 31.9%

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

   11. Applied egg-rr 31.9%

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

       1. pow1/2 31.9%

          \[\leadsto -\color{blue}{{\left(\frac{F \cdot 2}{B}\right)}^{0.5}} \]

       2. associate-/l* 31.9%

          \[\leadsto -{\color{blue}{\left(F \cdot \frac{2}{B}\right)}}^{0.5} \]

       3. unpow-prod-down 44.0%

          \[\leadsto -\color{blue}{{F}^{0.5} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]

       4. pow1/2 44.0%

          \[\leadsto -\color{blue}{\sqrt{F}} \cdot {\left(\frac{2}{B}\right)}^{0.5} \]

   13. Applied egg-rr 44.0%

       \[\leadsto -\color{blue}{\sqrt{F} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]
   14. Step-by-step derivation

       1. unpow1/2 44.0%

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

   15. Simplified 44.0%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-1}{A \cdot \frac{C \cdot -4}{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)\right) \cdot \left(4 \cdot C - \frac{{B}^{2}}{A}\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := F \cdot t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{t\_1 \cdot \left(4 \cdot C\right)}}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_1} \cdot \sqrt{2 \cdot C}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))) (t_1 (* F t_0)))
   (if (<= (pow B_m 2.0) 5e-176)
     (/ -1.0 (/ t_0 (sqrt (* t_1 (* 4.0 C)))))
     (if (<= (pow B_m 2.0) 4e+221)
       (/ (* (sqrt (* 2.0 t_1)) (sqrt (* 2.0 C))) (- t_0))
       (* (sqrt F) (- (sqrt (/ 2.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = F * t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 5e-176) {
		tmp = -1.0 / (t_0 / sqrt((t_1 * (4.0 * C))));
	} else if (pow(B_m, 2.0) <= 4e+221) {
		tmp = (sqrt((2.0 * t_1)) * sqrt((2.0 * C))) / -t_0;
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(F * t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-176)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(t_1 * Float64(4.0 * C)))));
	elseif ((B_m ^ 2.0) <= 4e+221)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_1)) * sqrt(Float64(2.0 * C))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * t$95$0), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-176], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(t$95$1 * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+221], N[(N[(N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e-176

   1. Initial program 19.7%

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

   3. Simplified 31.3%

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

   5. Taylor expanded in A around -inf 23.1%

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

      1. *-commutative 23.1%

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

   7. Simplified 23.1%

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

      1. clear-num 23.1%

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

      2. inv-pow 23.1%

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

      3. *-commutative 23.1%

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

      4. *-commutative 23.1%

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

      5. *-commutative 23.1%

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

   9. Applied egg-rr 23.1%

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

       1. unpow-1 23.1%

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

   11. Simplified 23.1%

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

   if 5e-176 < (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000002e221

   1. Initial program 34.5%

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

   3. Simplified 40.7%

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

      1. associate-*r* 40.7%

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

      2. associate-+r+ 40.0%

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

      3. hypot-undefine 34.5%

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

      4. unpow2 34.5%

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

      5. unpow2 34.5%

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

      6. +-commutative 34.5%

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

      7. sqrt-prod 38.1%

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

      8. *-commutative 38.1%

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

      9. associate-+l+ 38.6%

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

   6. Applied egg-rr 52.8%

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

   7. Taylor expanded in A around -inf 28.5%

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

   if 4.0000000000000002e221 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 2.9%

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

   3. Taylor expanded in B around inf 31.6%

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

      1. mul-1-neg 31.6%

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

      2. *-commutative 31.6%

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

   5. Simplified 31.6%

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

      1. *-commutative 31.6%

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

      2. pow1/2 31.6%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 31.6%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 31.9%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 31.9%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 31.9%

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

   9. Simplified 31.9%

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

       1. associate-*l/ 31.9%

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

   11. Applied egg-rr 31.9%

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

       1. pow1/2 31.9%

          \[\leadsto -\color{blue}{{\left(\frac{F \cdot 2}{B}\right)}^{0.5}} \]

       2. associate-/l* 31.9%

          \[\leadsto -{\color{blue}{\left(F \cdot \frac{2}{B}\right)}}^{0.5} \]

       3. unpow-prod-down 44.0%

          \[\leadsto -\color{blue}{{F}^{0.5} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]

       4. pow1/2 44.0%

          \[\leadsto -\color{blue}{\sqrt{F}} \cdot {\left(\frac{2}{B}\right)}^{0.5} \]

   13. Applied egg-rr 44.0%

       \[\leadsto -\color{blue}{\sqrt{F} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]
   14. Step-by-step derivation

       1. unpow1/2 44.0%

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

   15. Simplified 44.0%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := F \cdot t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{t\_1 \cdot \left(4 \cdot C\right)}}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot \sqrt{4 \cdot C}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))) (t_1 (* F t_0)))
   (if (<= (pow B_m 2.0) 5e-176)
     (/ -1.0 (/ t_0 (sqrt (* t_1 (* 4.0 C)))))
     (if (<= (pow B_m 2.0) 4e+221)
       (/ (* (sqrt t_1) (sqrt (* 4.0 C))) (- t_0))
       (* (sqrt F) (- (sqrt (/ 2.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = F * t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 5e-176) {
		tmp = -1.0 / (t_0 / sqrt((t_1 * (4.0 * C))));
	} else if (pow(B_m, 2.0) <= 4e+221) {
		tmp = (sqrt(t_1) * sqrt((4.0 * C))) / -t_0;
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(F * t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-176)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(t_1 * Float64(4.0 * C)))));
	elseif ((B_m ^ 2.0) <= 4e+221)
		tmp = Float64(Float64(sqrt(t_1) * sqrt(Float64(4.0 * C))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * t$95$0), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-176], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(t$95$1 * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+221], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[N[(4.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e-176

   1. Initial program 19.7%

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

   3. Simplified 31.3%

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

   5. Taylor expanded in A around -inf 23.1%

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

      1. *-commutative 23.1%

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

   7. Simplified 23.1%

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

      1. clear-num 23.1%

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

      2. inv-pow 23.1%

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

      3. *-commutative 23.1%

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

      4. *-commutative 23.1%

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

      5. *-commutative 23.1%

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

   9. Applied egg-rr 23.1%

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

       1. unpow-1 23.1%

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

   11. Simplified 23.1%

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

   if 5e-176 < (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000002e221

   1. Initial program 34.5%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in A around -inf 26.1%

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

      1. *-commutative 26.1%

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

   7. Simplified 26.1%

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

      1. sqrt-prod 28.5%

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

      2. *-commutative 28.5%

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

      3. *-commutative 28.5%

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

   9. Applied egg-rr 28.5%

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

   if 4.0000000000000002e221 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 2.9%

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

   3. Taylor expanded in B around inf 31.6%

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

      1. mul-1-neg 31.6%

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

      2. *-commutative 31.6%

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

   5. Simplified 31.6%

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

      1. *-commutative 31.6%

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

      2. pow1/2 31.6%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 31.6%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 31.9%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 31.9%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 31.9%

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

   9. Simplified 31.9%

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

       1. associate-*l/ 31.9%

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

   11. Applied egg-rr 31.9%

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

       1. pow1/2 31.9%

          \[\leadsto -\color{blue}{{\left(\frac{F \cdot 2}{B}\right)}^{0.5}} \]

       2. associate-/l* 31.9%

          \[\leadsto -{\color{blue}{\left(F \cdot \frac{2}{B}\right)}}^{0.5} \]

       3. unpow-prod-down 44.0%

          \[\leadsto -\color{blue}{{F}^{0.5} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]

       4. pow1/2 44.0%

          \[\leadsto -\color{blue}{\sqrt{F}} \cdot {\left(\frac{2}{B}\right)}^{0.5} \]

   13. Applied egg-rr 44.0%

       \[\leadsto -\color{blue}{\sqrt{F} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]
   14. Step-by-step derivation

       1. unpow1/2 44.0%

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

   15. Simplified 44.0%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{4 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-75}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-25}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-75)
   (/
    (sqrt (* (* F (fma B_m B_m (* A (* C -4.0)))) (* 4.0 C)))
    (* (* 4.0 A) C))
   (if (<= (pow B_m 2.0) 1e-25)
     (- (sqrt (/ F (- A))))
     (if (<= (pow B_m 2.0) 5e+294)
       (* (sqrt (* F (+ C (hypot C B_m)))) (/ (sqrt 2.0) (- B_m)))
       (* (sqrt F) (- (sqrt (/ 2.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-75) {
		tmp = sqrt(((F * fma(B_m, B_m, (A * (C * -4.0)))) * (4.0 * C))) / ((4.0 * A) * C);
	} else if (pow(B_m, 2.0) <= 1e-25) {
		tmp = -sqrt((F / -A));
	} else if (pow(B_m, 2.0) <= 5e+294) {
		tmp = sqrt((F * (C + hypot(C, B_m)))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-75)
		tmp = Float64(sqrt(Float64(Float64(F * fma(B_m, B_m, Float64(A * Float64(C * -4.0)))) * Float64(4.0 * C))) / Float64(Float64(4.0 * A) * C));
	elseif ((B_m ^ 2.0) <= 1e-25)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	elseif ((B_m ^ 2.0) <= 5e+294)
		tmp = Float64(sqrt(Float64(F * Float64(C + hypot(C, B_m)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-75], N[(N[Sqrt[N[(N[(F * N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-25], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+294], N[(N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999996e-76

   1. Initial program 22.5%

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

   3. Simplified 32.8%

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

   5. Taylor expanded in A around -inf 25.5%

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

      1. *-commutative 25.5%

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

   7. Simplified 25.5%

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

   8. Taylor expanded in B around 0 23.7%

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

      1. associate-*r* 23.7%

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

   10. Simplified 23.7%

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

   if 9.9999999999999996e-76 < (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000004e-25

   1. Initial program 23.9%

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

   3. Simplified 32.8%

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

   5. Taylor expanded in A around -inf 28.5%

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

   6. Taylor expanded in F around 0 37.3%

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

   7. Taylor expanded in B around 0 55.2%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 55.2%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]

      2. neg-mul-1 55.2%

         \[\leadsto -1 \cdot \sqrt{\frac{\color{blue}{-F}}{A}} \]

   9. Simplified 55.2%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if 1.00000000000000004e-25 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999999e294

   1. Initial program 33.2%

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

   3. Taylor expanded in A around 0 8.7%

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

      1. mul-1-neg 8.7%

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

      2. *-commutative 8.7%

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

      3. *-commutative 8.7%

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

      4. +-commutative 8.7%

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

      5. unpow2 8.7%

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

      6. unpow2 8.7%

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

      7. hypot-define 12.4%

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

   5. Simplified 12.4%

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

   if 4.9999999999999999e294 < (pow.f64 B #s(literal 2 binary64))

   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 33.6%

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

      1. mul-1-neg 33.6%

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

      2. *-commutative 33.6%

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

   5. Simplified 33.6%

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

      1. *-commutative 33.6%

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

      2. pow1/2 33.6%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 33.6%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 33.8%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 33.8%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 33.8%

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

   9. Simplified 33.8%

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

       1. associate-*l/ 33.8%

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

   11. Applied egg-rr 33.8%

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

       1. pow1/2 33.8%

          \[\leadsto -\color{blue}{{\left(\frac{F \cdot 2}{B}\right)}^{0.5}} \]

       2. associate-/l* 33.8%

          \[\leadsto -{\color{blue}{\left(F \cdot \frac{2}{B}\right)}}^{0.5} \]

       3. unpow-prod-down 47.7%

          \[\leadsto -\color{blue}{{F}^{0.5} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]

       4. pow1/2 47.7%

          \[\leadsto -\color{blue}{\sqrt{F}} \cdot {\left(\frac{2}{B}\right)}^{0.5} \]

   13. Applied egg-rr 47.7%

       \[\leadsto -\color{blue}{\sqrt{F} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]
   14. Step-by-step derivation

       1. unpow1/2 47.7%

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

   15. Simplified 47.7%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-75}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-25}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(4 \cdot C\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 4e+221)
     (/ (sqrt (* (* F t_0) (* 4.0 C))) (- t_0))
     (* (sqrt F) (- (sqrt (/ 2.0 B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 4e+221) {
		tmp = sqrt(((F * t_0) * (4.0 * C))) / -t_0;
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e+221)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(4.0 * C))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+221], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000002e221

   1. Initial program 25.9%

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

   3. Simplified 35.3%

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

   5. Taylor expanded in A around -inf 24.4%

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

      1. *-commutative 24.4%

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

   7. Simplified 24.4%

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

   if 4.0000000000000002e221 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 2.9%

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

   3. Taylor expanded in B around inf 31.6%

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

      1. mul-1-neg 31.6%

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

      2. *-commutative 31.6%

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

   5. Simplified 31.6%

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

      1. *-commutative 31.6%

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

      2. pow1/2 31.6%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 31.6%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 31.9%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 31.9%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 31.9%

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

   9. Simplified 31.9%

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

       1. associate-*l/ 31.9%

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

   11. Applied egg-rr 31.9%

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

       1. pow1/2 31.9%

          \[\leadsto -\color{blue}{{\left(\frac{F \cdot 2}{B}\right)}^{0.5}} \]

       2. associate-/l* 31.9%

          \[\leadsto -{\color{blue}{\left(F \cdot \frac{2}{B}\right)}}^{0.5} \]

       3. unpow-prod-down 44.0%

          \[\leadsto -\color{blue}{{F}^{0.5} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]

       4. pow1/2 44.0%

          \[\leadsto -\color{blue}{\sqrt{F}} \cdot {\left(\frac{2}{B}\right)}^{0.5} \]

   13. Applied egg-rr 44.0%

       \[\leadsto -\color{blue}{\sqrt{F} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]
   14. Step-by-step derivation

       1. unpow1/2 44.0%

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

   15. Simplified 44.0%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-75}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+221}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-75)
   (/
    (sqrt (* (* F (fma B_m B_m (* A (* C -4.0)))) (* 4.0 C)))
    (* (* 4.0 A) C))
   (if (<= (pow B_m 2.0) 4e+221)
     (- (sqrt (/ F (- A))))
     (* (sqrt F) (- (sqrt (/ 2.0 B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-75) {
		tmp = sqrt(((F * fma(B_m, B_m, (A * (C * -4.0)))) * (4.0 * C))) / ((4.0 * A) * C);
	} else if (pow(B_m, 2.0) <= 4e+221) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-75)
		tmp = Float64(sqrt(Float64(Float64(F * fma(B_m, B_m, Float64(A * Float64(C * -4.0)))) * Float64(4.0 * C))) / Float64(Float64(4.0 * A) * C));
	elseif ((B_m ^ 2.0) <= 4e+221)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-75], N[(N[Sqrt[N[(N[(F * N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+221], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999996e-76

   1. Initial program 22.5%

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

   3. Simplified 32.8%

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

   5. Taylor expanded in A around -inf 25.5%

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

      1. *-commutative 25.5%

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

   7. Simplified 25.5%

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

   8. Taylor expanded in B around 0 23.7%

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

      1. associate-*r* 23.7%

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

   10. Simplified 23.7%

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

   if 9.9999999999999996e-76 < (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000002e221

   1. Initial program 32.4%

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

   3. Simplified 39.9%

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

   5. Taylor expanded in A around -inf 25.2%

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

   6. Taylor expanded in F around 0 19.5%

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

   7. Taylor expanded in B around 0 22.6%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 22.6%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]

      2. neg-mul-1 22.6%

         \[\leadsto -1 \cdot \sqrt{\frac{\color{blue}{-F}}{A}} \]

   9. Simplified 22.6%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if 4.0000000000000002e221 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 2.9%

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

   3. Taylor expanded in B around inf 31.6%

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

      1. mul-1-neg 31.6%

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

      2. *-commutative 31.6%

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

   5. Simplified 31.6%

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

      1. *-commutative 31.6%

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

      2. pow1/2 31.6%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 31.6%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 31.9%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 31.9%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 31.9%

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

   9. Simplified 31.9%

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

       1. associate-*l/ 31.9%

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

   11. Applied egg-rr 31.9%

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

       1. pow1/2 31.9%

          \[\leadsto -\color{blue}{{\left(\frac{F \cdot 2}{B}\right)}^{0.5}} \]

       2. associate-/l* 31.9%

          \[\leadsto -{\color{blue}{\left(F \cdot \frac{2}{B}\right)}}^{0.5} \]

       3. unpow-prod-down 44.0%

          \[\leadsto -\color{blue}{{F}^{0.5} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]

       4. pow1/2 44.0%

          \[\leadsto -\color{blue}{\sqrt{F}} \cdot {\left(\frac{2}{B}\right)}^{0.5} \]

   13. Applied egg-rr 44.0%

       \[\leadsto -\color{blue}{\sqrt{F} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]
   14. Step-by-step derivation

       1. unpow1/2 44.0%

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

   15. Simplified 44.0%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-75}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+221}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{+221}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 4e+221)
   (- (sqrt (/ F (- A))))
   (* (sqrt F) (- (sqrt (/ 2.0 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 4e+221) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	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) <= 4d+221) then
        tmp = -sqrt((f / -a))
    else
        tmp = sqrt(f) * -sqrt((2.0d0 / b_m))
    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) <= 4e+221) {
		tmp = -Math.sqrt((F / -A));
	} else {
		tmp = Math.sqrt(F) * -Math.sqrt((2.0 / B_m));
	}
	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) <= 4e+221:
		tmp = -math.sqrt((F / -A))
	else:
		tmp = math.sqrt(F) * -math.sqrt((2.0 / B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e+221)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	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) <= 4e+221)
		tmp = -sqrt((F / -A));
	else
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	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], 4e+221], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000002e221

   1. Initial program 25.9%

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

   3. Simplified 35.3%

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

   5. Taylor expanded in A around -inf 24.4%

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

   6. Taylor expanded in F around 0 16.0%

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

   7. Taylor expanded in B around 0 20.5%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 20.5%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]

      2. neg-mul-1 20.5%

         \[\leadsto -1 \cdot \sqrt{\frac{\color{blue}{-F}}{A}} \]

   9. Simplified 20.5%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if 4.0000000000000002e221 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 2.9%

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

   3. Taylor expanded in B around inf 31.6%

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

      1. mul-1-neg 31.6%

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

      2. *-commutative 31.6%

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

   5. Simplified 31.6%

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

      1. *-commutative 31.6%

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

      2. pow1/2 31.6%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 31.6%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 31.9%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 31.9%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 31.9%

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

   9. Simplified 31.9%

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

       1. associate-*l/ 31.9%

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

   11. Applied egg-rr 31.9%

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

       1. pow1/2 31.9%

          \[\leadsto -\color{blue}{{\left(\frac{F \cdot 2}{B}\right)}^{0.5}} \]

       2. associate-/l* 31.9%

          \[\leadsto -{\color{blue}{\left(F \cdot \frac{2}{B}\right)}}^{0.5} \]

       3. unpow-prod-down 44.0%

          \[\leadsto -\color{blue}{{F}^{0.5} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]

       4. pow1/2 44.0%

          \[\leadsto -\color{blue}{\sqrt{F}} \cdot {\left(\frac{2}{B}\right)}^{0.5} \]

   13. Applied egg-rr 44.0%

       \[\leadsto -\color{blue}{\sqrt{F} \cdot {\left(\frac{2}{B}\right)}^{0.5}} \]
   14. Step-by-step derivation

       1. unpow1/2 44.0%

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

   15. Simplified 44.0%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{+221}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7.5 \cdot 10^{+109}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \left|\frac{F}{B\_m}\right|}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7.5e+109)
   (- (sqrt (/ F (- A))))
   (- (sqrt (* 2.0 (fabs (/ 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) {
	double tmp;
	if (B_m <= 7.5e+109) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = -sqrt((2.0 * fabs((F / B_m))));
	}
	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 <= 7.5d+109) then
        tmp = -sqrt((f / -a))
    else
        tmp = -sqrt((2.0d0 * abs((f / b_m))))
    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 (B_m <= 7.5e+109) {
		tmp = -Math.sqrt((F / -A));
	} else {
		tmp = -Math.sqrt((2.0 * Math.abs((F / B_m))));
	}
	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 B_m <= 7.5e+109:
		tmp = -math.sqrt((F / -A))
	else:
		tmp = -math.sqrt((2.0 * math.fabs((F / B_m))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7.5e+109)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(-sqrt(Float64(2.0 * abs(Float64(F / B_m)))));
	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 <= 7.5e+109)
		tmp = -sqrt((F / -A));
	else
		tmp = -sqrt((2.0 * abs((F / B_m))));
	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[B$95$m, 7.5e+109], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(2.0 * N[Abs[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 7.50000000000000018e109

   1. Initial program 23.1%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in A around -inf 20.8%

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

   6. Taylor expanded in F around 0 13.8%

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

   7. Taylor expanded in B around 0 18.5%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 18.5%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]

      2. neg-mul-1 18.5%

         \[\leadsto -1 \cdot \sqrt{\frac{\color{blue}{-F}}{A}} \]

   9. Simplified 18.5%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if 7.50000000000000018e109 < B

   1. Initial program 0.2%

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

   3. Taylor expanded in B around inf 52.8%

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

      1. mul-1-neg 52.8%

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

      2. *-commutative 52.8%

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

   5. Simplified 52.8%

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

      1. *-commutative 52.8%

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

      2. pow1/2 52.8%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 52.8%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 53.2%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 53.2%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 53.2%

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

   9. Simplified 53.2%

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

       1. add-sqr-sqrt 52.8%

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

       2. pow1/2 52.8%

          \[\leadsto -\sqrt{\left(\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{\frac{F}{B}}\right) \cdot 2} \]

       3. pow1/2 52.8%

          \[\leadsto -\sqrt{\left({\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}}\right) \cdot 2} \]

       4. pow-prod-down 38.6%

          \[\leadsto -\sqrt{\color{blue}{{\left(\frac{F}{B} \cdot \frac{F}{B}\right)}^{0.5}} \cdot 2} \]

       5. pow2 38.6%

          \[\leadsto -\sqrt{{\color{blue}{\left({\left(\frac{F}{B}\right)}^{2}\right)}}^{0.5} \cdot 2} \]

   11. Applied egg-rr 38.6%

       \[\leadsto -\sqrt{\color{blue}{{\left({\left(\frac{F}{B}\right)}^{2}\right)}^{0.5}} \cdot 2} \]
   12. Step-by-step derivation

       1. unpow1/2 38.6%

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

       2. unpow2 38.6%

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

       3. rem-sqrt-square 53.2%

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

   13. Simplified 53.2%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.5 \cdot 10^{+109}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \left|\frac{F}{B}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.8 \cdot 10^{+110}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5.8e+110) (- (sqrt (/ F (- A)))) (- (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) {
	double tmp;
	if (B_m <= 5.8e+110) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = -sqrt((F * (2.0 / B_m)));
	}
	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 <= 5.8d+110) then
        tmp = -sqrt((f / -a))
    else
        tmp = -sqrt((f * (2.0d0 / b_m)))
    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 (B_m <= 5.8e+110) {
		tmp = -Math.sqrt((F / -A));
	} else {
		tmp = -Math.sqrt((F * (2.0 / B_m)));
	}
	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 B_m <= 5.8e+110:
		tmp = -math.sqrt((F / -A))
	else:
		tmp = -math.sqrt((F * (2.0 / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5.8e+110)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(-sqrt(Float64(F * Float64(2.0 / B_m))));
	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 <= 5.8e+110)
		tmp = -sqrt((F / -A));
	else
		tmp = -sqrt((F * (2.0 / B_m)));
	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[B$95$m, 5.8e+110], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 5.7999999999999999e110

   1. Initial program 23.1%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in A around -inf 20.8%

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

   6. Taylor expanded in F around 0 13.8%

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

   7. Taylor expanded in B around 0 18.5%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 18.5%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]

      2. neg-mul-1 18.5%

         \[\leadsto -1 \cdot \sqrt{\frac{\color{blue}{-F}}{A}} \]

   9. Simplified 18.5%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if 5.7999999999999999e110 < B

   1. Initial program 0.2%

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

   3. Taylor expanded in B around inf 52.8%

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

      1. mul-1-neg 52.8%

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

      2. *-commutative 52.8%

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

   5. Simplified 52.8%

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

      1. *-commutative 52.8%

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

      2. pow1/2 52.8%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

      3. pow1/2 52.8%

         \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

      4. pow-prod-down 53.2%

         \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

   7. Applied egg-rr 53.2%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 53.2%

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

   9. Simplified 53.2%

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

       1. associate-*l/ 53.2%

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

   11. Applied egg-rr 53.2%

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

       1. associate-/l* 53.2%

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

   13. Applied egg-rr 53.2%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.8 \cdot 10^{+110}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 25.5% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.3%

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

3. Taylor expanded in B around inf 11.3%

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

   1. mul-1-neg 11.3%

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

   2. *-commutative 11.3%

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

5. Simplified 11.3%

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

   1. *-commutative 11.3%

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

   2. pow1/2 11.5%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2} \]

   3. pow1/2 11.5%

      \[\leadsto -{\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}} \]

   4. pow-prod-down 11.5%

      \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]

7. Applied egg-rr 11.5%

   \[\leadsto -\color{blue}{{\left(\frac{F}{B} \cdot 2\right)}^{0.5}} \]
8. Step-by-step derivation

   1. unpow1/2 11.4%

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

9. Simplified 11.4%

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

    1. associate-*l/ 11.4%

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

11. Applied egg-rr 11.4%

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

    1. associate-/l* 11.4%

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

13. Applied egg-rr 11.4%

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

Reproduce

herbie shell --seed 2024180 
(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))))