ABCF->ab-angle a

Percentage Accurate: 18.0% → 59.6%

Time: 25.8s

Alternatives: 11

Speedup: 6.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.0% 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: 59.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := -\sqrt{F}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot t\_1\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t\_0\right)} \cdot \sqrt{-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B\_m}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (- (sqrt F)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 -2e-180)
     (*
      (sqrt 2.0)
      (*
       (sqrt
        (/ (+ (+ A C) (hypot B_m (- A C))) (fma -4.0 (* A C) (pow B_m 2.0))))
       t_1))
     (if (<= t_3 INFINITY)
       (/
        (*
         (sqrt (* 2.0 (* F t_0)))
         (sqrt (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
        (- t_0))
       (* (sqrt (/ 2.0 B_m)) t_1)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = -sqrt(F);
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -2e-180) {
		tmp = sqrt(2.0) * (sqrt((((A + C) + hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0)))) * t_1);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * (F * t_0))) * sqrt(((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / -t_0;
	} else {
		tmp = sqrt((2.0 / B_m)) * t_1;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(-sqrt(F))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -2e-180)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))) * t_1));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * sqrt(Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(2.0 / B_m)) * t_1);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[F], $MachinePrecision])}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-180], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[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] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2e-180

   1. Initial program 34.5%

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

   3. Taylor expanded in F around 0 38.7%

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

      1. mul-1-neg 38.7%

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

      2. *-commutative 38.7%

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

      3. cancel-sign-sub-inv 38.7%

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

      4. metadata-eval 38.7%

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

      5. +-commutative 38.7%

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

   5. Simplified 56.2%

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

      1. pow1/2 56.2%

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

      2. associate-/l* 65.8%

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

      3. unpow-prod-down 77.1%

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

      4. pow1/2 77.1%

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

      5. associate-+r+ 76.1%

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

      6. *-commutative 76.1%

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

   7. Applied egg-rr 76.1%

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

      1. unpow1/2 76.1%

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

   9. Simplified 76.1%

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

   if -2e-180 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 10.8%

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

   3. Simplified 20.7%

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

      1. associate-*r* 20.7%

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

      2. associate-+r+ 19.4%

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

      3. hypot-undefine 10.8%

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

      4. unpow2 10.8%

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

      5. unpow2 10.8%

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

      6. +-commutative 10.8%

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

      7. sqrt-prod 16.6%

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

      8. *-commutative 16.6%

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

      9. associate-+l+ 17.3%

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

   6. Applied egg-rr 35.8%

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

   7. Taylor expanded in A around -inf 31.2%

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

   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 14.6%

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

      1. mul-1-neg 14.6%

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

      2. *-commutative 14.6%

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

   5. Simplified 14.6%

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

      1. *-commutative 14.6%

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

      2. pow1/2 14.6%

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

      3. pow1/2 14.6%

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

      4. pow-prod-down 14.6%

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

   7. Applied egg-rr 14.6%

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

      1. unpow1/2 14.6%

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

   9. Simplified 14.6%

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

   10. Taylor expanded in F around 0 14.6%

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

       1. associate-*r/ 14.6%

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

       2. *-commutative 14.6%

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

       3. associate-/l* 14.6%

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

   12. Simplified 14.6%

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

       1. pow1/2 14.6%

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

       2. *-commutative 14.6%

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

       3. unpow-prod-down 20.5%

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

       4. pow1/2 20.5%

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

       5. pow1/2 20.5%

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

   14. Applied egg-rr 20.5%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 53.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;B\_m \leq 1.7 \cdot 10^{-190}:\\ \;\;\;\;-\sqrt{2 \cdot \left(\left(F \cdot -0.5\right) \cdot \frac{1}{A}\right)}\\ \mathbf{elif}\;B\_m \leq 1.6 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B\_m \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{F \cdot \left(2 \cdot t\_1\right)} \cdot \frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)}}{-t\_1}\\ \mathbf{elif}\;B\_m \leq 1.02 \cdot 10^{+154}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F} \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma -4.0 (* A C) (pow B_m 2.0)))
        (t_1 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= B_m 1.7e-190)
     (- (sqrt (* 2.0 (* (* F -0.5) (/ 1.0 A)))))
     (if (<= B_m 1.6e-125)
       (/ (sqrt (* t_0 (* F (* 4.0 C)))) (- (fma B_m B_m (* A (* C -4.0)))))
       (if (<= B_m 1.35e+90)
         (*
          (sqrt (* F (* 2.0 t_1)))
          (/ (sqrt (+ (+ A C) (hypot (- A C) B_m))) (- t_1)))
         (if (<= B_m 1.02e+154)
           (- (sqrt (* 2.0 (* F (/ (+ (+ A C) (hypot B_m (- A 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(-4.0, (A * C), pow(B_m, 2.0));
	double t_1 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (B_m <= 1.7e-190) {
		tmp = -sqrt((2.0 * ((F * -0.5) * (1.0 / A))));
	} else if (B_m <= 1.6e-125) {
		tmp = sqrt((t_0 * (F * (4.0 * C)))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else if (B_m <= 1.35e+90) {
		tmp = sqrt((F * (2.0 * t_1))) * (sqrt(((A + C) + hypot((A - C), B_m))) / -t_1);
	} else if (B_m <= 1.02e+154) {
		tmp = -sqrt((2.0 * (F * (((A + C) + hypot(B_m, (A - 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(-4.0, Float64(A * C), (B_m ^ 2.0))
	t_1 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B_m <= 1.7e-190)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(Float64(F * -0.5) * Float64(1.0 / A)))));
	elseif (B_m <= 1.6e-125)
		tmp = Float64(sqrt(Float64(t_0 * Float64(F * Float64(4.0 * C)))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	elseif (B_m <= 1.35e+90)
		tmp = Float64(sqrt(Float64(F * Float64(2.0 * t_1))) * Float64(sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m))) / Float64(-t_1)));
	elseif (B_m <= 1.02e+154)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / t_0)))));
	else
		tmp = Float64(-Float64(sqrt(F) * 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[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.7e-190], (-N[Sqrt[N[(2.0 * N[(N[(F * -0.5), $MachinePrecision] * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[B$95$m, 1.6e-125], N[(N[Sqrt[N[(t$95$0 * N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 1.35e+90], N[(N[Sqrt[N[(F * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.02e+154], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / t$95$0), $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 B < 1.69999999999999991e-190

   1. Initial program 15.8%

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

   3. Taylor expanded in F around 0 19.2%

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

      1. mul-1-neg 19.2%

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

      2. *-commutative 19.2%

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

      3. cancel-sign-sub-inv 19.2%

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

      4. metadata-eval 19.2%

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

      5. +-commutative 19.2%

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

   5. Simplified 27.7%

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

   6. Taylor expanded in A around -inf 16.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. sqrt-unprod 16.3%

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

      2. associate-*r/ 16.3%

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{-0.5 \cdot F}{A}}} \]

   8. Applied egg-rr 16.3%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot \frac{-0.5 \cdot F}{A}}} \]
   9. Step-by-step derivation

      1. div-inv 16.3%

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

   10. Applied egg-rr 16.3%

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

   if 1.69999999999999991e-190 < B < 1.5999999999999999e-125

   1. Initial program 8.1%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in A around -inf 47.5%

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

      1. *-commutative 47.5%

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

   7. Simplified 47.5%

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

   8. Taylor expanded in F around 0 47.5%

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

      1. associate-*r* 47.5%

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

      2. *-commutative 47.5%

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

      3. +-commutative 47.5%

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

      4. unpow2 47.5%

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

      5. *-commutative 47.5%

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

      6. associate-*r* 47.5%

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

      7. fma-undefine 47.5%

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

      8. associate-*r* 53.9%

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

      9. *-commutative 53.9%

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

      10. *-commutative 53.9%

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

      11. fma-undefine 53.9%

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

      12. unpow2 53.9%

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

      13. associate-*r* 53.9%

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

      14. *-commutative 53.9%

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

      15. +-commutative 53.9%

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

      16. fma-define 53.9%

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

   10. Simplified 53.9%

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

   if 1.5999999999999999e-125 < B < 1.35e90

   1. Initial program 22.1%

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

   3. Simplified 35.1%

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

      1. associate-*r* 35.1%

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

      2. associate-+r+ 34.1%

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

      3. hypot-undefine 22.1%

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

      4. unpow2 22.1%

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

      5. unpow2 22.1%

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

      6. +-commutative 22.1%

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

      7. sqrt-prod 28.8%

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

      8. *-commutative 28.8%

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

      9. associate-+l+ 29.1%

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

   6. Applied egg-rr 48.1%

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

      1. associate-/l* 48.2%

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

      2. associate-*l* 48.2%

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

      3. associate-*r* 48.2%

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

      4. associate-+r+ 47.5%

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

      5. associate-*r* 47.5%

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

   8. Applied egg-rr 47.5%

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

   if 1.35e90 < B < 1.02000000000000007e154

   1. Initial program 20.8%

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

   3. Taylor expanded in F around 0 39.0%

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

      1. mul-1-neg 39.0%

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

      2. *-commutative 39.0%

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

      3. cancel-sign-sub-inv 39.0%

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

      4. metadata-eval 39.0%

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

      5. +-commutative 39.0%

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

   5. Simplified 45.9%

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

      1. *-commutative 45.9%

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

      2. pow1/2 45.9%

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

      3. pow1/2 45.9%

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

      4. pow-prod-down 45.8%

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

      5. associate-/l* 57.7%

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

      6. associate-+r+ 57.7%

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

      7. *-commutative 57.7%

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

   7. Applied egg-rr 57.7%

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

      1. unpow1/2 57.7%

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

   9. Simplified 57.7%

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

   if 1.02000000000000007e154 < B

   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 43.6%

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

      1. mul-1-neg 43.6%

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

      2. *-commutative 43.6%

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

   5. Simplified 43.6%

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

      1. *-commutative 43.6%

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

      2. pow1/2 43.6%

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

      3. pow1/2 43.6%

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

      4. pow-prod-down 43.8%

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

   7. Applied egg-rr 43.8%

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

      1. unpow1/2 43.8%

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

   9. Simplified 43.8%

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

   10. Taylor expanded in F around 0 43.8%

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

       1. associate-*r/ 43.8%

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

       2. *-commutative 43.8%

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

       3. associate-/l* 43.8%

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

   12. Simplified 43.8%

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

       1. pow1/2 43.8%

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

       2. *-commutative 43.8%

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

       3. unpow-prod-down 66.0%

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

       4. pow1/2 66.0%

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

       5. pow1/2 66.0%

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

   14. Applied egg-rr 66.0%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.7 \cdot 10^{-190}:\\ \;\;\;\;-\sqrt{2 \cdot \left(\left(F \cdot -0.5\right) \cdot \frac{1}{A}\right)}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{+154}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F} \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 8.8 \cdot 10^{-219}:\\ \;\;\;\;-\sqrt{2 \cdot \left(\left(F \cdot -0.5\right) \cdot \frac{1}{A}\right)}\\ \mathbf{elif}\;B\_m \leq 1.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t\_0\right)} \cdot \sqrt{-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 2.5 \cdot 10^{+153}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F} \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 8.8e-219)
     (- (sqrt (* 2.0 (* (* F -0.5) (/ 1.0 A)))))
     (if (<= B_m 1.8e-61)
       (/
        (*
         (sqrt (* 2.0 (* F t_0)))
         (sqrt (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
        (- t_0))
       (if (<= B_m 2.5e+153)
         (-
          (sqrt
           (*
            2.0
            (*
             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 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 8.8e-219) {
		tmp = -sqrt((2.0 * ((F * -0.5) * (1.0 / A))));
	} else if (B_m <= 1.8e-61) {
		tmp = (sqrt((2.0 * (F * t_0))) * sqrt(((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / -t_0;
	} else if (B_m <= 2.5e+153) {
		tmp = -sqrt((2.0 * (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 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 8.8e-219)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(Float64(F * -0.5) * Float64(1.0 / A)))));
	elseif (B_m <= 1.8e-61)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * sqrt(Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / Float64(-t_0));
	elseif (B_m <= 2.5e+153)
		tmp = Float64(-sqrt(Float64(2.0 * 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(-Float64(sqrt(F) * 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[B$95$m, 8.8e-219], (-N[Sqrt[N[(2.0 * N[(N[(F * -0.5), $MachinePrecision] * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[B$95$m, 1.8e-61], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 2.5e+153], (-N[Sqrt[N[(2.0 * 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 4 regimes

2. if B < 8.7999999999999998e-219

   1. Initial program 15.3%

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

   3. Taylor expanded in F around 0 19.5%

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

      1. mul-1-neg 19.5%

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

      2. *-commutative 19.5%

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

      3. cancel-sign-sub-inv 19.5%

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

      4. metadata-eval 19.5%

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

      5. +-commutative 19.5%

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

   5. Simplified 28.5%

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

   6. Taylor expanded in A around -inf 16.5%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. sqrt-unprod 16.5%

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

      2. associate-*r/ 16.5%

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{-0.5 \cdot F}{A}}} \]

   8. Applied egg-rr 16.5%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot \frac{-0.5 \cdot F}{A}}} \]
   9. Step-by-step derivation

      1. div-inv 16.5%

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

   10. Applied egg-rr 16.5%

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

   if 8.7999999999999998e-219 < B < 1.80000000000000007e-61

   1. Initial program 16.0%

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

   3. Simplified 27.2%

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

      1. associate-*r* 27.2%

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

      2. associate-+r+ 25.6%

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

      3. hypot-undefine 16.0%

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

      4. unpow2 16.0%

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

      5. unpow2 16.0%

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

      6. +-commutative 16.0%

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

      7. sqrt-prod 26.1%

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

      8. *-commutative 26.1%

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

      9. associate-+l+ 26.9%

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

   6. Applied egg-rr 45.3%

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

   7. Taylor expanded in A around -inf 38.6%

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

   if 1.80000000000000007e-61 < B < 2.50000000000000009e153

   1. Initial program 23.0%

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

   3. Taylor expanded in F around 0 25.2%

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

      1. mul-1-neg 25.2%

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

      2. *-commutative 25.2%

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

      3. cancel-sign-sub-inv 25.2%

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

      4. metadata-eval 25.2%

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

      5. +-commutative 25.2%

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

   5. Simplified 31.2%

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

      1. *-commutative 31.2%

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

      2. pow1/2 31.2%

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

      3. pow1/2 31.2%

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

      4. pow-prod-down 31.3%

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

      5. associate-/l* 41.2%

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

      6. associate-+r+ 40.8%

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

      7. *-commutative 40.8%

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

   7. Applied egg-rr 40.8%

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

      1. unpow1/2 40.8%

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

   9. Simplified 40.8%

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

   if 2.50000000000000009e153 < B

   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 43.6%

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

      1. mul-1-neg 43.6%

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

      2. *-commutative 43.6%

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

   5. Simplified 43.6%

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

      1. *-commutative 43.6%

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

      2. pow1/2 43.6%

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

      3. pow1/2 43.6%

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

      4. pow-prod-down 43.8%

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

   7. Applied egg-rr 43.8%

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

      1. unpow1/2 43.8%

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

   9. Simplified 43.8%

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

   10. Taylor expanded in F around 0 43.8%

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

       1. associate-*r/ 43.8%

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

       2. *-commutative 43.8%

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

       3. associate-/l* 43.8%

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

   12. Simplified 43.8%

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

       1. pow1/2 43.8%

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

       2. *-commutative 43.8%

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

       3. unpow-prod-down 66.0%

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

       4. pow1/2 66.0%

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

       5. pow1/2 66.0%

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

   14. Applied egg-rr 66.0%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.8 \cdot 10^{-219}:\\ \;\;\;\;-\sqrt{2 \cdot \left(\left(F \cdot -0.5\right) \cdot \frac{1}{A}\right)}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{+153}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F} \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)\\ \mathbf{if}\;B\_m \leq 8.5 \cdot 10^{-191}:\\ \;\;\;\;-\sqrt{2 \cdot \left(\left(F \cdot -0.5\right) \cdot \frac{1}{A}\right)}\\ \mathbf{elif}\;B\_m \leq 5.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B\_m \leq 1.25 \cdot 10^{-57}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F \cdot -0.5}{A}}\\ \mathbf{elif}\;B\_m \leq 6.1 \cdot 10^{+153}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F} \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma -4.0 (* A C) (pow B_m 2.0))))
   (if (<= B_m 8.5e-191)
     (- (sqrt (* 2.0 (* (* F -0.5) (/ 1.0 A)))))
     (if (<= B_m 5.5e-85)
       (/ (sqrt (* t_0 (* F (* 4.0 C)))) (- (fma B_m B_m (* A (* C -4.0)))))
       (if (<= B_m 1.25e-57)
         (- (sqrt (* 2.0 (/ (* F -0.5) A))))
         (if (<= B_m 6.1e+153)
           (- (sqrt (* 2.0 (* F (/ (+ (+ A C) (hypot B_m (- A 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(-4.0, (A * C), pow(B_m, 2.0));
	double tmp;
	if (B_m <= 8.5e-191) {
		tmp = -sqrt((2.0 * ((F * -0.5) * (1.0 / A))));
	} else if (B_m <= 5.5e-85) {
		tmp = sqrt((t_0 * (F * (4.0 * C)))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else if (B_m <= 1.25e-57) {
		tmp = -sqrt((2.0 * ((F * -0.5) / A)));
	} else if (B_m <= 6.1e+153) {
		tmp = -sqrt((2.0 * (F * (((A + C) + hypot(B_m, (A - 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(-4.0, Float64(A * C), (B_m ^ 2.0))
	tmp = 0.0
	if (B_m <= 8.5e-191)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(Float64(F * -0.5) * Float64(1.0 / A)))));
	elseif (B_m <= 5.5e-85)
		tmp = Float64(sqrt(Float64(t_0 * Float64(F * Float64(4.0 * C)))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	elseif (B_m <= 1.25e-57)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(Float64(F * -0.5) / A))));
	elseif (B_m <= 6.1e+153)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / t_0)))));
	else
		tmp = Float64(-Float64(sqrt(F) * 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[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 8.5e-191], (-N[Sqrt[N[(2.0 * N[(N[(F * -0.5), $MachinePrecision] * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[B$95$m, 5.5e-85], N[(N[Sqrt[N[(t$95$0 * N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 1.25e-57], (-N[Sqrt[N[(2.0 * N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[B$95$m, 6.1e+153], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / t$95$0), $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 B < 8.49999999999999954e-191

   1. Initial program 15.8%

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

   3. Taylor expanded in F around 0 19.2%

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

      1. mul-1-neg 19.2%

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

      2. *-commutative 19.2%

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

      3. cancel-sign-sub-inv 19.2%

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

      4. metadata-eval 19.2%

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

      5. +-commutative 19.2%

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

   5. Simplified 27.7%

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

   6. Taylor expanded in A around -inf 16.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. sqrt-unprod 16.3%

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

      2. associate-*r/ 16.3%

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{-0.5 \cdot F}{A}}} \]

   8. Applied egg-rr 16.3%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot \frac{-0.5 \cdot F}{A}}} \]
   9. Step-by-step derivation

      1. div-inv 16.3%

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

   10. Applied egg-rr 16.3%

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

   if 8.49999999999999954e-191 < B < 5.4999999999999997e-85

   1. Initial program 9.8%

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

   3. Simplified 23.9%

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

   5. Taylor expanded in A around -inf 41.3%

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

      1. *-commutative 41.3%

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

   7. Simplified 41.3%

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

   8. Taylor expanded in F around 0 41.3%

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

      1. associate-*r* 41.3%

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

      2. *-commutative 41.3%

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

      3. +-commutative 41.3%

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

      4. unpow2 41.3%

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

      5. *-commutative 41.3%

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

      6. associate-*r* 41.3%

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

      7. fma-undefine 41.3%

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

      8. associate-*r* 45.3%

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

      9. *-commutative 45.3%

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

      10. *-commutative 45.3%

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

      11. fma-undefine 45.3%

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

      12. unpow2 45.3%

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

      13. associate-*r* 45.3%

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

      14. *-commutative 45.3%

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

      15. +-commutative 45.3%

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

      16. fma-define 45.3%

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

   10. Simplified 45.3%

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

   if 5.4999999999999997e-85 < B < 1.25e-57

   1. Initial program 24.7%

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

   3. Taylor expanded in F around 0 24.2%

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

      1. mul-1-neg 24.2%

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

      2. *-commutative 24.2%

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

      3. cancel-sign-sub-inv 24.2%

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

      4. metadata-eval 24.2%

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

      5. +-commutative 24.2%

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

   5. Simplified 36.9%

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

   6. Taylor expanded in A around -inf 34.5%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. sqrt-unprod 34.9%

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

      2. associate-*r/ 34.9%

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{-0.5 \cdot F}{A}}} \]

   8. Applied egg-rr 34.9%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot \frac{-0.5 \cdot F}{A}}} \]

   if 1.25e-57 < B < 6.0999999999999998e153

   1. Initial program 23.5%

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

   3. Taylor expanded in F around 0 25.7%

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

      1. mul-1-neg 25.7%

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

      2. *-commutative 25.7%

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

      3. cancel-sign-sub-inv 25.7%

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

      4. metadata-eval 25.7%

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

      5. +-commutative 25.7%

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

   5. Simplified 31.9%

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

      1. *-commutative 31.9%

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

      2. pow1/2 31.9%

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

      3. pow1/2 31.9%

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

      4. pow-prod-down 32.0%

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

      5. associate-/l* 39.7%

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

      6. associate-+r+ 39.3%

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

      7. *-commutative 39.3%

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

   7. Applied egg-rr 39.3%

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

      1. unpow1/2 39.3%

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

   9. Simplified 39.3%

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

   if 6.0999999999999998e153 < B

   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 43.6%

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

      1. mul-1-neg 43.6%

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

      2. *-commutative 43.6%

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

   5. Simplified 43.6%

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

      1. *-commutative 43.6%

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

      2. pow1/2 43.6%

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

      3. pow1/2 43.6%

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

      4. pow-prod-down 43.8%

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

   7. Applied egg-rr 43.8%

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

      1. unpow1/2 43.8%

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

   9. Simplified 43.8%

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

   10. Taylor expanded in F around 0 43.8%

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

       1. associate-*r/ 43.8%

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

       2. *-commutative 43.8%

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

       3. associate-/l* 43.8%

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

   12. Simplified 43.8%

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

       1. pow1/2 43.8%

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

       2. *-commutative 43.8%

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

       3. unpow-prod-down 66.0%

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

       4. pow1/2 66.0%

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

       5. pow1/2 66.0%

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

   14. Applied egg-rr 66.0%

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

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.5 \cdot 10^{-191}:\\ \;\;\;\;-\sqrt{2 \cdot \left(\left(F \cdot -0.5\right) \cdot \frac{1}{A}\right)}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-57}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F \cdot -0.5}{A}}\\ \mathbf{elif}\;B \leq 6.1 \cdot 10^{+153}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F} \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 9 \cdot 10^{-219}:\\ \;\;\;\;-\sqrt{2 \cdot \left(\left(F \cdot -0.5\right) \cdot \frac{1}{A}\right)}\\ \mathbf{elif}\;B\_m \leq 1.02 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t\_0\right)} \cdot \sqrt{2 \cdot C}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 4.4 \cdot 10^{+153}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F} \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 9e-219)
     (- (sqrt (* 2.0 (* (* F -0.5) (/ 1.0 A)))))
     (if (<= B_m 1.02e-39)
       (/ (* (sqrt (* 2.0 (* F t_0))) (sqrt (* 2.0 C))) (- t_0))
       (if (<= B_m 4.4e+153)
         (-
          (sqrt
           (*
            2.0
            (*
             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 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 9e-219) {
		tmp = -sqrt((2.0 * ((F * -0.5) * (1.0 / A))));
	} else if (B_m <= 1.02e-39) {
		tmp = (sqrt((2.0 * (F * t_0))) * sqrt((2.0 * C))) / -t_0;
	} else if (B_m <= 4.4e+153) {
		tmp = -sqrt((2.0 * (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 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 9e-219)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(Float64(F * -0.5) * Float64(1.0 / A)))));
	elseif (B_m <= 1.02e-39)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * sqrt(Float64(2.0 * C))) / Float64(-t_0));
	elseif (B_m <= 4.4e+153)
		tmp = Float64(-sqrt(Float64(2.0 * 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(-Float64(sqrt(F) * 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[B$95$m, 9e-219], (-N[Sqrt[N[(2.0 * N[(N[(F * -0.5), $MachinePrecision] * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[B$95$m, 1.02e-39], 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], If[LessEqual[B$95$m, 4.4e+153], (-N[Sqrt[N[(2.0 * 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 4 regimes

2. if B < 9.00000000000000029e-219

   1. Initial program 15.3%

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

   3. Taylor expanded in F around 0 19.5%

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

      1. mul-1-neg 19.5%

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

      2. *-commutative 19.5%

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

      3. cancel-sign-sub-inv 19.5%

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

      4. metadata-eval 19.5%

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

      5. +-commutative 19.5%

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

   5. Simplified 28.5%

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

   6. Taylor expanded in A around -inf 16.5%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. sqrt-unprod 16.5%

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

      2. associate-*r/ 16.5%

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{-0.5 \cdot F}{A}}} \]

   8. Applied egg-rr 16.5%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot \frac{-0.5 \cdot F}{A}}} \]
   9. Step-by-step derivation

      1. div-inv 16.5%

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

   10. Applied egg-rr 16.5%

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

   if 9.00000000000000029e-219 < B < 1.02000000000000007e-39

   1. Initial program 17.0%

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

   3. Simplified 29.3%

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

      1. associate-*r* 29.3%

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

      2. associate-+r+ 27.8%

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

      3. hypot-undefine 17.0%

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

      4. unpow2 17.0%

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

      5. unpow2 17.0%

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

      6. +-commutative 17.0%

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

      7. sqrt-prod 26.1%

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

      8. *-commutative 26.1%

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

      9. associate-+l+ 26.9%

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

   6. Applied egg-rr 45.7%

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

   7. Taylor expanded in A around -inf 37.4%

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

   if 1.02000000000000007e-39 < B < 4.3999999999999999e153

   1. Initial program 22.6%

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

   3. Taylor expanded in F around 0 27.7%

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

      1. mul-1-neg 27.7%

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

      2. *-commutative 27.7%

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

      3. cancel-sign-sub-inv 27.7%

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

      4. metadata-eval 27.7%

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

      5. +-commutative 27.7%

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

   5. Simplified 34.4%

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

      1. *-commutative 34.4%

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

      2. pow1/2 34.4%

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

      3. pow1/2 34.4%

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

      4. pow-prod-down 34.5%

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

      5. associate-/l* 42.7%

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

      6. associate-+r+ 42.3%

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

      7. *-commutative 42.3%

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

   7. Applied egg-rr 42.3%

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

      1. unpow1/2 42.3%

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

   9. Simplified 42.3%

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

   if 4.3999999999999999e153 < B

   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 43.6%

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

      1. mul-1-neg 43.6%

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

      2. *-commutative 43.6%

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

   5. Simplified 43.6%

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

      1. *-commutative 43.6%

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

      2. pow1/2 43.6%

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

      3. pow1/2 43.6%

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

      4. pow-prod-down 43.8%

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

   7. Applied egg-rr 43.8%

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

      1. unpow1/2 43.8%

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

   9. Simplified 43.8%

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

   10. Taylor expanded in F around 0 43.8%

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

       1. associate-*r/ 43.8%

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

       2. *-commutative 43.8%

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

       3. associate-/l* 43.8%

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

   12. Simplified 43.8%

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

       1. pow1/2 43.8%

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

       2. *-commutative 43.8%

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

       3. unpow-prod-down 66.0%

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

       4. pow1/2 66.0%

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

       5. pow1/2 66.0%

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

   14. Applied egg-rr 66.0%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-219}:\\ \;\;\;\;-\sqrt{2 \cdot \left(\left(F \cdot -0.5\right) \cdot \frac{1}{A}\right)}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{+153}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F} \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.16 \cdot 10^{-57}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B\_m \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B\_m}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

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 1.16e-57)
   (- (sqrt (/ (- F) A)))
   (if (<= B_m 7.6e+153)
     (-
      (sqrt
       (*
        2.0
        (*
         F
         (/
          (+ (+ A C) (hypot B_m (- A C)))
          (fma -4.0 (* A C) (pow B_m 2.0)))))))
     (* (sqrt (/ 2.0 B_m)) (- (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.16e-57) {
		tmp = -sqrt((-F / A));
	} else if (B_m <= 7.6e+153) {
		tmp = -sqrt((2.0 * (F * (((A + C) + hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0))))));
	} else {
		tmp = sqrt((2.0 / B_m)) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.16e-57)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	elseif (B_m <= 7.6e+153)
		tmp = Float64(-sqrt(Float64(2.0 * 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(Float64(2.0 / B_m)) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.16e-57], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), If[LessEqual[B$95$m, 7.6e+153], (-N[Sqrt[N[(2.0 * 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[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.15999999999999996e-57

   1. Initial program 15.4%

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

   3. Taylor expanded in F around 0 18.2%

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

      1. mul-1-neg 18.2%

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

      2. *-commutative 18.2%

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

      3. cancel-sign-sub-inv 18.2%

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

      4. metadata-eval 18.2%

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

      5. +-commutative 18.2%

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

   5. Simplified 26.5%

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

   6. Taylor expanded in A around -inf 17.8%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. sqrt-unprod 17.9%

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

      2. associate-*r/ 17.9%

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{-0.5 \cdot F}{A}}} \]

   8. Applied egg-rr 17.9%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot \frac{-0.5 \cdot F}{A}}} \]

   9. Taylor expanded in F around 0 17.9%

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

       1. mul-1-neg 17.9%

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

   11. Simplified 17.9%

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

   if 1.15999999999999996e-57 < B < 7.59999999999999933e153

   1. Initial program 23.5%

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

   3. Taylor expanded in F around 0 25.7%

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

      1. mul-1-neg 25.7%

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

      2. *-commutative 25.7%

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

      3. cancel-sign-sub-inv 25.7%

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

      4. metadata-eval 25.7%

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

      5. +-commutative 25.7%

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

   5. Simplified 31.9%

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

      1. *-commutative 31.9%

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

      2. pow1/2 31.9%

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

      3. pow1/2 31.9%

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

      4. pow-prod-down 32.0%

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

      5. associate-/l* 39.7%

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

      6. associate-+r+ 39.3%

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

      7. *-commutative 39.3%

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

   7. Applied egg-rr 39.3%

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

      1. unpow1/2 39.3%

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

   9. Simplified 39.3%

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

   if 7.59999999999999933e153 < B

   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 43.6%

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

      1. mul-1-neg 43.6%

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

      2. *-commutative 43.6%

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

   5. Simplified 43.6%

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

      1. *-commutative 43.6%

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

      2. pow1/2 43.6%

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

      3. pow1/2 43.6%

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

      4. pow-prod-down 43.8%

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

   7. Applied egg-rr 43.8%

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

      1. unpow1/2 43.8%

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

   9. Simplified 43.8%

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

   10. Taylor expanded in F around 0 43.8%

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

       1. associate-*r/ 43.8%

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

       2. *-commutative 43.8%

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

       3. associate-/l* 43.8%

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

   12. Simplified 43.8%

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

       1. pow1/2 43.8%

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

       2. *-commutative 43.8%

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

       3. unpow-prod-down 66.0%

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

       4. pow1/2 66.0%

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

       5. pow1/2 66.0%

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

   14. Applied egg-rr 66.0%

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

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.16 \cdot 10^{-57}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.5 \cdot 10^{+95}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B\_m}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

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 3.5e+95)
   (- (sqrt (/ (- F) A)))
   (* (sqrt (/ 2.0 B_m)) (- (sqrt F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.5e+95) {
		tmp = -sqrt((-F / A));
	} else {
		tmp = sqrt((2.0 / B_m)) * -sqrt(F);
	}
	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 <= 3.5d+95) then
        tmp = -sqrt((-f / a))
    else
        tmp = sqrt((2.0d0 / b_m)) * -sqrt(f)
    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 <= 3.5e+95) {
		tmp = -Math.sqrt((-F / A));
	} else {
		tmp = Math.sqrt((2.0 / B_m)) * -Math.sqrt(F);
	}
	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 <= 3.5e+95:
		tmp = -math.sqrt((-F / A))
	else:
		tmp = math.sqrt((2.0 / B_m)) * -math.sqrt(F)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.5e+95)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	else
		tmp = Float64(sqrt(Float64(2.0 / B_m)) * Float64(-sqrt(F)));
	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 <= 3.5e+95)
		tmp = -sqrt((-F / A));
	else
		tmp = sqrt((2.0 / B_m)) * -sqrt(F);
	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, 3.5e+95], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.5e95

   1. Initial program 16.4%

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

   3. Taylor expanded in F around 0 17.9%

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

      1. mul-1-neg 17.9%

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

      2. *-commutative 17.9%

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

      3. cancel-sign-sub-inv 17.9%

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

      4. metadata-eval 17.9%

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

      5. +-commutative 17.9%

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

   5. Simplified 25.9%

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

   6. Taylor expanded in A around -inf 18.5%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. sqrt-unprod 18.6%

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

      2. associate-*r/ 18.6%

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{-0.5 \cdot F}{A}}} \]

   8. Applied egg-rr 18.6%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot \frac{-0.5 \cdot F}{A}}} \]

   9. Taylor expanded in F around 0 18.6%

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

       1. mul-1-neg 18.6%

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

   11. Simplified 18.6%

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

   if 3.5e95 < B

   1. Initial program 7.4%

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

   3. Taylor expanded in B around inf 45.7%

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

      1. mul-1-neg 45.7%

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

      2. *-commutative 45.7%

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

   5. Simplified 45.7%

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

      1. *-commutative 45.7%

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

      2. pow1/2 45.7%

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

      3. pow1/2 45.7%

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

      4. pow-prod-down 45.8%

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

   7. Applied egg-rr 45.8%

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

      1. unpow1/2 45.8%

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

   9. Simplified 45.8%

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

   10. Taylor expanded in F around 0 45.8%

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

       1. associate-*r/ 45.8%

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

       2. *-commutative 45.8%

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

       3. associate-/l* 45.8%

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

   12. Simplified 45.8%

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

       1. pow1/2 45.8%

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

       2. *-commutative 45.8%

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

       3. unpow-prod-down 62.7%

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

       4. pow1/2 62.7%

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

       5. pow1/2 62.7%

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

   14. Applied egg-rr 62.7%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.5 \cdot 10^{+95}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4 \cdot 10^{+95}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{1}{\frac{B\_m}{F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4e+95)
   (- (sqrt (/ (- F) A)))
   (- (sqrt (* 2.0 (/ 1.0 (/ B_m F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4e+95) {
		tmp = -sqrt((-F / A));
	} else {
		tmp = -sqrt((2.0 * (1.0 / (B_m / F))));
	}
	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 <= 4d+95) then
        tmp = -sqrt((-f / a))
    else
        tmp = -sqrt((2.0d0 * (1.0d0 / (b_m / f))))
    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 <= 4e+95) {
		tmp = -Math.sqrt((-F / A));
	} else {
		tmp = -Math.sqrt((2.0 * (1.0 / (B_m / F))));
	}
	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 <= 4e+95:
		tmp = -math.sqrt((-F / A))
	else:
		tmp = -math.sqrt((2.0 * (1.0 / (B_m / F))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4e+95)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	else
		tmp = Float64(-sqrt(Float64(2.0 * Float64(1.0 / Float64(B_m / F)))));
	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 <= 4e+95)
		tmp = -sqrt((-F / A));
	else
		tmp = -sqrt((2.0 * (1.0 / (B_m / F))));
	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, 4e+95], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(2.0 * N[(1.0 / N[(B$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 4.00000000000000008e95

   1. Initial program 16.4%

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

   3. Taylor expanded in F around 0 17.9%

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

      1. mul-1-neg 17.9%

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

      2. *-commutative 17.9%

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

      3. cancel-sign-sub-inv 17.9%

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

      4. metadata-eval 17.9%

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

      5. +-commutative 17.9%

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

   5. Simplified 25.9%

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

   6. Taylor expanded in A around -inf 18.5%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. sqrt-unprod 18.6%

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

      2. associate-*r/ 18.6%

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{-0.5 \cdot F}{A}}} \]

   8. Applied egg-rr 18.6%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot \frac{-0.5 \cdot F}{A}}} \]

   9. Taylor expanded in F around 0 18.6%

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

       1. mul-1-neg 18.6%

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

   11. Simplified 18.6%

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

   if 4.00000000000000008e95 < B

   1. Initial program 7.4%

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

   3. Taylor expanded in B around inf 45.7%

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

      1. mul-1-neg 45.7%

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

      2. *-commutative 45.7%

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

   5. Simplified 45.7%

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

      1. *-commutative 45.7%

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

      2. pow1/2 45.7%

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

      3. pow1/2 45.7%

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

      4. pow-prod-down 45.8%

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

   7. Applied egg-rr 45.8%

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

      1. unpow1/2 45.8%

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

   9. Simplified 45.8%

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

       1. clear-num 45.8%

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

       2. inv-pow 45.8%

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

   11. Applied egg-rr 45.8%

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

       1. unpow-1 45.8%

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

   13. Simplified 45.8%

       \[\leadsto -\sqrt{\color{blue}{\frac{1}{\frac{B}{F}}} \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4 \cdot 10^{+95}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{1}{\frac{B}{F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.1 \cdot 10^{+95}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

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 4.1e+95) (- (sqrt (/ (- F) A))) (- (sqrt (* 2.0 (/ F B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.1e+95) {
		tmp = -sqrt((-F / A));
	} else {
		tmp = -sqrt((2.0 * (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 <= 4.1d+95) then
        tmp = -sqrt((-f / a))
    else
        tmp = -sqrt((2.0d0 * (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 <= 4.1e+95) {
		tmp = -Math.sqrt((-F / A));
	} else {
		tmp = -Math.sqrt((2.0 * (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 <= 4.1e+95:
		tmp = -math.sqrt((-F / A))
	else:
		tmp = -math.sqrt((2.0 * (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 <= 4.1e+95)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	else
		tmp = Float64(-sqrt(Float64(2.0 * 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 <= 4.1e+95)
		tmp = -sqrt((-F / A));
	else
		tmp = -sqrt((2.0 * (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, 4.1e+95], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 4.09999999999999986e95

   1. Initial program 16.4%

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

   3. Taylor expanded in F around 0 17.9%

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

      1. mul-1-neg 17.9%

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

      2. *-commutative 17.9%

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

      3. cancel-sign-sub-inv 17.9%

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

      4. metadata-eval 17.9%

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

      5. +-commutative 17.9%

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

   5. Simplified 25.9%

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

   6. Taylor expanded in A around -inf 18.5%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. sqrt-unprod 18.6%

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

      2. associate-*r/ 18.6%

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{-0.5 \cdot F}{A}}} \]

   8. Applied egg-rr 18.6%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot \frac{-0.5 \cdot F}{A}}} \]

   9. Taylor expanded in F around 0 18.6%

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

       1. mul-1-neg 18.6%

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

   11. Simplified 18.6%

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

   if 4.09999999999999986e95 < B

   1. Initial program 7.4%

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

   3. Taylor expanded in B around inf 45.7%

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

      1. mul-1-neg 45.7%

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

      2. *-commutative 45.7%

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

   5. Simplified 45.7%

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

      1. *-commutative 45.7%

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

      2. pow1/2 45.7%

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

      3. pow1/2 45.7%

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

      4. pow-prod-down 45.8%

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

   7. Applied egg-rr 45.8%

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

      1. unpow1/2 45.8%

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

   9. Simplified 45.8%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.1 \cdot 10^{+95}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3 \cdot 10^{+95}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B\_m}}\\ \end{array} \end{array} \]

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 3e+95) (- (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 <= 3e+95) {
		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 <= 3d+95) 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 <= 3e+95) {
		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 <= 3e+95:
		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 <= 3e+95)
		tmp = Float64(-sqrt(Float64(Float64(-F) / 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 <= 3e+95)
		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, 3e+95], (-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 < 2.99999999999999991e95

   1. Initial program 16.4%

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

   3. Taylor expanded in F around 0 17.9%

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

      1. mul-1-neg 17.9%

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

      2. *-commutative 17.9%

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

      3. cancel-sign-sub-inv 17.9%

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

      4. metadata-eval 17.9%

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

      5. +-commutative 17.9%

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

   5. Simplified 25.9%

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

   6. Taylor expanded in A around -inf 18.5%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. sqrt-unprod 18.6%

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

      2. associate-*r/ 18.6%

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{-0.5 \cdot F}{A}}} \]

   8. Applied egg-rr 18.6%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot \frac{-0.5 \cdot F}{A}}} \]

   9. Taylor expanded in F around 0 18.6%

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

       1. mul-1-neg 18.6%

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

   11. Simplified 18.6%

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

   if 2.99999999999999991e95 < B

   1. Initial program 7.4%

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

   3. Taylor expanded in B around inf 45.7%

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

      1. mul-1-neg 45.7%

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

      2. *-commutative 45.7%

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

   5. Simplified 45.7%

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

      1. *-commutative 45.7%

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

      2. pow1/2 45.7%

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

      3. pow1/2 45.7%

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

      4. pow-prod-down 45.8%

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

   7. Applied egg-rr 45.8%

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

      1. unpow1/2 45.8%

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

   9. Simplified 45.8%

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

   10. Taylor expanded in F around 0 45.8%

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

       1. associate-*r/ 45.8%

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

       2. *-commutative 45.8%

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

       3. associate-/l* 45.8%

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

   12. Simplified 45.8%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3 \cdot 10^{+95}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.0% accurate, 6.0× speedup

Math

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

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) A))))

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 / A));
}

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 / a))
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 / A));
}

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 / A))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(Float64(-F) / A)))
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 / A));
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) / A), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 14.8%

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

3. Taylor expanded in F around 0 17.2%

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

   1. mul-1-neg 17.2%

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

   2. *-commutative 17.2%

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

   3. cancel-sign-sub-inv 17.2%

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

   4. metadata-eval 17.2%

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

   5. +-commutative 17.2%

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

5. Simplified 24.6%

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

6. Taylor expanded in A around -inf 16.7%

   \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
7. Step-by-step derivation

   1. sqrt-unprod 16.8%

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

   2. associate-*r/ 16.8%

      \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{-0.5 \cdot F}{A}}} \]

8. Applied egg-rr 16.8%

   \[\leadsto -\color{blue}{\sqrt{2 \cdot \frac{-0.5 \cdot F}{A}}} \]

9. Taylor expanded in F around 0 16.9%

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

    1. mul-1-neg 16.9%

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

11. Simplified 16.9%

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

12. Final simplification 16.9%

    \[\leadsto -\sqrt{\frac{-F}{A}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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))))