ABCF->ab-angle a

Percentage Accurate: 18.7% → 61.3%

Time: 34.0s

Alternatives: 17

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: 18.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 61.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_2 := t\_0 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{-187}:\\ \;\;\;\;\left(\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(A \cdot C, -4, {B\_m}^{2}\right)}} \cdot \sqrt{F}\right) \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq 10^{-71}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_2 (- t_0 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt (* t_1 (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2)))
   (if (<= t_3 -4e-187)
     (*
      (*
       (sqrt
        (/ (+ (+ A C) (hypot B_m (- A C))) (fma (* A C) -4.0 (pow B_m 2.0))))
       (sqrt F))
      (- (sqrt 2.0)))
     (if (<= t_3 1e-71)
       (/ (sqrt (* t_1 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C)))) t_2)
       (if (<= t_3 INFINITY)
         (/
          (*
           (sqrt (* 2.0 (* F (fma B_m B_m (* (* A C) -4.0)))))
           (sqrt (+ A (+ C (hypot (- A C) B_m)))))
          (- (fma B_m B_m (* A (* C -4.0)))))
         (/ (sqrt (* 2.0 F)) (- (sqrt 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 = (4.0 * A) * C;
	double t_1 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_2 = t_0 - pow(B_m, 2.0);
	double t_3 = sqrt((t_1 * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -4e-187) {
		tmp = (sqrt((((A + C) + hypot(B_m, (A - C))) / fma((A * C), -4.0, pow(B_m, 2.0)))) * sqrt(F)) * -sqrt(2.0);
	} else if (t_3 <= 1e-71) {
		tmp = sqrt((t_1 * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * (F * fma(B_m, B_m, ((A * C) * -4.0))))) * sqrt((A + (C + hypot((A - C), B_m))))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	tmp = 0.0
	if (t_3 <= -4e-187)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(Float64(A * C), -4.0, (B_m ^ 2.0)))) * sqrt(F)) * Float64(-sqrt(2.0)));
	elseif (t_3 <= 1e-71)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * fma(B_m, B_m, Float64(Float64(A * C) * -4.0))))) * sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m))))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -4e-187], N[(N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[(A * C), $MachinePrecision] * -4.0 + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, 1e-71], N[(N[Sqrt[N[(t$95$1 * 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$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * B$95$m + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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))) < -4.0000000000000001e-187

   1. Initial program 42.6%

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

   3. Taylor expanded in F around 0 43.8%

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

      1. mul-1-neg 43.8%

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

      2. *-commutative 43.8%

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

      3. distribute-rgt-neg-in 43.8%

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

      4. associate-/l* 46.9%

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

      5. cancel-sign-sub-inv 46.9%

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

      6. metadata-eval 46.9%

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

      7. +-commutative 46.9%

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

   5. Simplified 71.4%

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

      1. pow1/2 71.4%

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

      2. *-commutative 71.4%

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

      3. unpow-prod-down 82.3%

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

      4. pow1/2 82.3%

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

      5. associate-+r+ 81.8%

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

      6. fma-undefine 81.8%

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

      7. *-commutative 81.8%

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

      8. *-commutative 81.8%

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

      9. fma-define 81.8%

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

      10. pow1/2 81.8%

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

   7. Applied egg-rr 81.8%

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

   if -4.0000000000000001e-187 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 9.9999999999999992e-72

   1. Initial program 7.7%

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

   3. Taylor expanded in A around -inf 43.9%

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

   if 9.9999999999999992e-72 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 27.7%

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

   3. Simplified 47.0%

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

      1. associate-*r* 47.0%

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

      2. associate-+r+ 47.0%

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

      3. hypot-undefine 27.7%

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

      4. unpow2 27.7%

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

      5. unpow2 27.7%

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

      6. +-commutative 27.7%

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

      7. sqrt-prod 27.7%

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

      8. *-commutative 27.7%

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

      9. associate-*r* 27.7%

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

      10. associate-+l+ 27.7%

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

   6. Applied egg-rr 75.6%

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

   if +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 18.1%

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

      1. mul-1-neg 18.1%

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

      2. distribute-rgt-neg-in 18.1%

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

   5. Simplified 18.1%

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

      1. distribute-rgt-neg-out 18.1%

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

      2. pow1/2 18.3%

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

      3. pow1/2 18.3%

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

      4. pow-prod-down 18.4%

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

   7. Applied egg-rr 18.4%

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

      1. unpow1/2 18.2%

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

      2. associate-*l/ 18.2%

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

      3. sqrt-div 26.0%

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

   9. Applied egg-rr 26.0%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{-187}:\\ \;\;\;\;\left(\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(A \cdot C, -4, {B}^{2}\right)}} \cdot \sqrt{F}\right) \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 10^{-71}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := -\sqrt{2}\\ t_2 := A \cdot \left(C \cdot -4\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-215}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, t\_2\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+196}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{{B\_m}^{2} + t\_2}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)) (t_1 (- (sqrt 2.0))) (t_2 (* A (* C -4.0))))
   (if (<= (pow B_m 2.0) 5e-215)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 2e+63)
       (/
        (*
         (sqrt (* 2.0 (* F (fma B_m B_m (* (* A C) -4.0)))))
         (sqrt (+ A (+ C (hypot (- A C) B_m)))))
        (- (fma B_m B_m t_2)))
       (if (<= (pow B_m 2.0) 2e+151)
         (* (sqrt (* F (/ -0.5 A))) t_1)
         (if (<= (pow B_m 2.0) 2e+196)
           (*
            (sqrt
             (* F (/ (+ A (+ C (hypot B_m (- A C)))) (+ (pow B_m 2.0) t_2))))
            t_1)
           (/ (sqrt (* 2.0 F)) (- (sqrt 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 = (4.0 * A) * C;
	double t_1 = -sqrt(2.0);
	double t_2 = A * (C * -4.0);
	double tmp;
	if (pow(B_m, 2.0) <= 5e-215) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+63) {
		tmp = (sqrt((2.0 * (F * fma(B_m, B_m, ((A * C) * -4.0))))) * sqrt((A + (C + hypot((A - C), B_m))))) / -fma(B_m, B_m, t_2);
	} else if (pow(B_m, 2.0) <= 2e+151) {
		tmp = sqrt((F * (-0.5 / A))) * t_1;
	} else if (pow(B_m, 2.0) <= 2e+196) {
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / (pow(B_m, 2.0) + t_2)))) * t_1;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(-sqrt(2.0))
	t_2 = Float64(A * Float64(C * -4.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-215)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+63)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * fma(B_m, B_m, Float64(Float64(A * C) * -4.0))))) * sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m))))) / Float64(-fma(B_m, B_m, t_2)));
	elseif ((B_m ^ 2.0) <= 2e+151)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * t_1);
	elseif ((B_m ^ 2.0) <= 2e+196)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / Float64((B_m ^ 2.0) + t_2)))) * t_1);
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$2 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-215], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+63], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * B$95$m + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(B$95$m * B$95$m + t$95$2), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+151], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+196], N[(N[Sqrt[N[(F * N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999956e-215

   1. Initial program 19.0%

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

   3. Taylor expanded in A around -inf 23.8%

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

   if 4.99999999999999956e-215 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000012e63

   1. Initial program 38.6%

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

   3. Simplified 45.5%

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

      1. associate-*r* 45.5%

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

      2. associate-+r+ 45.2%

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

      3. hypot-undefine 38.6%

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

      4. unpow2 38.6%

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

      5. unpow2 38.6%

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

      6. +-commutative 38.6%

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

      7. sqrt-prod 43.2%

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

      8. *-commutative 43.2%

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

      9. associate-*r* 43.2%

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

      10. associate-+l+ 43.2%

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

   6. Applied egg-rr 63.7%

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

   if 2.00000000000000012e63 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000003e151

   1. Initial program 17.8%

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

   3. Taylor expanded in F around 0 17.5%

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

      1. mul-1-neg 17.5%

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

      2. *-commutative 17.5%

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

      3. distribute-rgt-neg-in 17.5%

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

      4. associate-/l* 22.5%

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

      5. cancel-sign-sub-inv 22.5%

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

      6. metadata-eval 22.5%

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

      7. +-commutative 22.5%

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

   5. Simplified 34.0%

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

   6. Taylor expanded in A around -inf 43.4%

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

   if 2.00000000000000003e151 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e196

   1. Initial program 24.9%

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

   3. Taylor expanded in F around 0 39.5%

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

      1. mul-1-neg 39.5%

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

      2. *-commutative 39.5%

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

      3. distribute-rgt-neg-in 39.5%

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

      4. associate-/l* 46.9%

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

      5. cancel-sign-sub-inv 46.9%

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

      6. metadata-eval 46.9%

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

      7. +-commutative 46.9%

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

   5. Simplified 76.9%

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

      1. fma-undefine 76.9%

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

      2. *-commutative 76.9%

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

      3. *-commutative 76.9%

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

      4. associate-*r* 76.9%

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

   7. Applied egg-rr 76.9%

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

   if 1.9999999999999999e196 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 2.8%

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

   3. Taylor expanded in B around inf 27.5%

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

      1. mul-1-neg 27.5%

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

      2. distribute-rgt-neg-in 27.5%

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

   5. Simplified 27.5%

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

      1. distribute-rgt-neg-out 27.5%

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

      2. pow1/2 27.5%

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

      3. pow1/2 27.5%

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

      4. pow-prod-down 27.7%

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

   7. Applied egg-rr 27.7%

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

      1. unpow1/2 27.7%

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

      2. associate-*l/ 27.7%

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

      3. sqrt-div 39.0%

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

   9. Applied egg-rr 39.0%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-215}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+196}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{{B}^{2} + A \cdot \left(C \cdot -4\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B\_m}^{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-68}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{\mathsf{fma}\left(A \cdot C, -4, {B\_m}^{2}\right)}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+143}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{B\_m \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)) (t_1 (- t_0 (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 1e-68)
     (/ (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C))) t_1)
     (if (<= (pow B_m 2.0) 2e+63)
       (-
        (sqrt
         (*
          2.0
          (/
           (* F (+ A (+ C (hypot B_m (- A C)))))
           (fma (* A C) -4.0 (pow B_m 2.0))))))
       (if (<= (pow B_m 2.0) 1e+143)
         (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
         (if (<= (pow B_m 2.0) 2e+305)
           (/ (* B_m (* (sqrt 2.0) (sqrt (* F (+ C (hypot B_m C)))))) t_1)
           (/ (sqrt (* 2.0 F)) (- (sqrt 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 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B_m, 2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 1e-68) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / t_1;
	} else if (pow(B_m, 2.0) <= 2e+63) {
		tmp = -sqrt((2.0 * ((F * (A + (C + hypot(B_m, (A - C))))) / fma((A * C), -4.0, pow(B_m, 2.0)))));
	} else if (pow(B_m, 2.0) <= 1e+143) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 2e+305) {
		tmp = (B_m * (sqrt(2.0) * sqrt((F * (C + hypot(B_m, C)))))) / t_1;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-68)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / t_1);
	elseif ((B_m ^ 2.0) <= 2e+63)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(Float64(F * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) / fma(Float64(A * C), -4.0, (B_m ^ 2.0))))));
	elseif ((B_m ^ 2.0) <= 1e+143)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 2e+305)
		tmp = Float64(Float64(B_m * Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + hypot(B_m, C)))))) / t_1);
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-68], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+63], (-N[Sqrt[N[(2.0 * N[(N[(F * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(A * C), $MachinePrecision] * -4.0 + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+143], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+305], N[(N[(B$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000007e-68

   1. Initial program 23.3%

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

   3. Taylor expanded in A around -inf 24.1%

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

   if 1.00000000000000007e-68 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000012e63

   1. Initial program 40.5%

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

   3. Taylor expanded in F around 0 45.6%

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

      1. mul-1-neg 45.6%

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

      2. *-commutative 45.6%

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

      3. distribute-rgt-neg-in 45.6%

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

      4. associate-/l* 45.7%

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

      5. cancel-sign-sub-inv 45.7%

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

      6. metadata-eval 45.7%

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

      7. +-commutative 45.7%

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

   5. Simplified 60.6%

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

      1. pow1 60.6%

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

   7. Applied egg-rr 59.6%

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

      1. unpow1 59.6%

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

      2. unpow1/2 59.6%

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

      3. associate-+l+ 60.5%

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

   9. Simplified 60.5%

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

   if 2.00000000000000012e63 < (pow.f64 B #s(literal 2 binary64)) < 1e143

   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 20.5%

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

      1. mul-1-neg 20.5%

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

      2. *-commutative 20.5%

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

      3. distribute-rgt-neg-in 20.5%

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

      4. associate-/l* 26.3%

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

      5. cancel-sign-sub-inv 26.3%

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

      6. metadata-eval 26.3%

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

      7. +-commutative 26.3%

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

   5. Simplified 33.5%

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

   6. Taylor expanded in A around -inf 44.6%

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

   if 1e143 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e305

   1. Initial program 17.1%

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

   3. Taylor expanded in A around 0 26.3%

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

      1. associate-*l* 26.3%

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

      2. unpow2 26.3%

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

      3. unpow2 26.3%

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

      4. hypot-define 32.4%

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

   5. Simplified 32.4%

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

   if 1.9999999999999999e305 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf 29.0%

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

      1. mul-1-neg 29.0%

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

      2. distribute-rgt-neg-in 29.0%

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

   5. Simplified 29.0%

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

      1. distribute-rgt-neg-out 29.0%

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

      2. pow1/2 29.0%

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

      3. pow1/2 29.0%

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

      4. pow-prod-down 29.2%

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

   7. Applied egg-rr 29.2%

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

      1. unpow1/2 29.2%

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

      2. associate-*l/ 29.2%

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

      3. sqrt-div 42.2%

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

   9. Applied egg-rr 42.2%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-68}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{\mathsf{fma}\left(A \cdot C, -4, {B}^{2}\right)}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+143}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{fma}\left(B\_m, B\_m, t\_1\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-215}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{F \cdot \left(2 \cdot t\_2\right)} \cdot \left(-\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)}}{t\_2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{{B\_m}^{2} + t\_1}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)) (t_1 (* A (* C -4.0))) (t_2 (fma B_m B_m t_1)))
   (if (<= (pow B_m 2.0) 5e-215)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 5e-35)
       (*
        (sqrt (* F (* 2.0 t_2)))
        (- (/ (sqrt (+ (+ A C) (hypot (- A C) B_m))) t_2)))
       (if (<= (pow B_m 2.0) 5e+276)
         (*
          (sqrt
           (* F (/ (+ A (+ C (hypot B_m (- A C)))) (+ (pow B_m 2.0) t_1))))
          (- (sqrt 2.0)))
         (/ (sqrt (* 2.0 F)) (- (sqrt 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 = (4.0 * A) * C;
	double t_1 = A * (C * -4.0);
	double t_2 = fma(B_m, B_m, t_1);
	double tmp;
	if (pow(B_m, 2.0) <= 5e-215) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 5e-35) {
		tmp = sqrt((F * (2.0 * t_2))) * -(sqrt(((A + C) + hypot((A - C), B_m))) / t_2);
	} else if (pow(B_m, 2.0) <= 5e+276) {
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / (pow(B_m, 2.0) + t_1)))) * -sqrt(2.0);
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(A * Float64(C * -4.0))
	t_2 = fma(B_m, B_m, t_1)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-215)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 5e-35)
		tmp = Float64(sqrt(Float64(F * Float64(2.0 * t_2))) * Float64(-Float64(sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m))) / t_2)));
	elseif ((B_m ^ 2.0) <= 5e+276)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / Float64((B_m ^ 2.0) + t_1)))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B$95$m * B$95$m + t$95$1), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-215], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-35], N[(N[Sqrt[N[(F * N[(2.0 * t$95$2), $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$2), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+276], N[(N[Sqrt[N[(F * N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999956e-215

   1. Initial program 19.0%

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

   3. Taylor expanded in A around -inf 23.8%

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

   if 4.99999999999999956e-215 < (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999964e-35

   1. Initial program 34.0%

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

   3. Simplified 40.0%

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

      1. associate-*r* 40.0%

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

      2. associate-+r+ 39.9%

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

      3. hypot-undefine 34.0%

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

      4. unpow2 34.0%

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

      5. unpow2 34.0%

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

      6. +-commutative 34.0%

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

      7. sqrt-prod 39.5%

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

      8. *-commutative 39.5%

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

      9. associate-*r* 39.5%

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

      10. associate-+l+ 39.5%

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

   6. Applied egg-rr 59.2%

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

      1. associate-/l* 59.1%

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

      2. associate-*l* 59.1%

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

      3. associate-*r* 59.1%

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

      4. associate-+r+ 59.0%

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

   8. Applied egg-rr 59.0%

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

   if 4.99999999999999964e-35 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000001e276

   1. Initial program 28.2%

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

   3. Taylor expanded in F around 0 36.2%

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

      1. mul-1-neg 36.2%

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

      2. *-commutative 36.2%

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

      3. distribute-rgt-neg-in 36.2%

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

      4. associate-/l* 40.1%

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

      5. cancel-sign-sub-inv 40.1%

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

      6. metadata-eval 40.1%

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

      7. +-commutative 40.1%

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

   5. Simplified 59.8%

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

      1. fma-undefine 59.8%

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

      2. *-commutative 59.8%

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

      3. *-commutative 59.8%

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

      4. associate-*r* 59.8%

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

   7. Applied egg-rr 59.8%

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

   if 5.00000000000000001e276 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.5%

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

   3. Taylor expanded in B around inf 28.8%

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

      1. mul-1-neg 28.8%

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

      2. distribute-rgt-neg-in 28.8%

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

   5. Simplified 28.8%

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

      1. distribute-rgt-neg-out 28.8%

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

      2. pow1/2 28.8%

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

      3. pow1/2 28.8%

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

      4. pow-prod-down 29.0%

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

   7. Applied egg-rr 29.0%

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

      1. unpow1/2 29.0%

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

      2. associate-*l/ 29.0%

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

      3. sqrt-div 41.0%

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

   9. Applied egg-rr 41.0%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-215}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{{B}^{2} + A \cdot \left(C \cdot -4\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-68}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{\mathsf{fma}\left(A \cdot C, -4, {B\_m}^{2}\right)}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{1}{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 1e-68)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 2e+63)
       (-
        (sqrt
         (*
          2.0
          (/
           (* F (+ A (+ C (hypot B_m (- A C)))))
           (fma (* A C) -4.0 (pow B_m 2.0))))))
       (if (<= (pow B_m 2.0) 5e+128)
         (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
         (if (<= (pow B_m 2.0) 2e+305)
           (/
            -1.0
            (* (/ B_m (sqrt 2.0)) (sqrt (/ 1.0 (* F (+ C (hypot B_m C)))))))
           (/ (sqrt (* 2.0 F)) (- (sqrt 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 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-68) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+63) {
		tmp = -sqrt((2.0 * ((F * (A + (C + hypot(B_m, (A - C))))) / fma((A * C), -4.0, pow(B_m, 2.0)))));
	} else if (pow(B_m, 2.0) <= 5e+128) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 2e+305) {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt((1.0 / (F * (C + hypot(B_m, C))))));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-68)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+63)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(Float64(F * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) / fma(Float64(A * C), -4.0, (B_m ^ 2.0))))));
	elseif ((B_m ^ 2.0) <= 5e+128)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 2e+305)
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(1.0 / Float64(F * Float64(C + hypot(B_m, C)))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-68], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+63], (-N[Sqrt[N[(2.0 * N[(N[(F * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(A * C), $MachinePrecision] * -4.0 + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+128], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+305], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000007e-68

   1. Initial program 23.3%

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

   3. Taylor expanded in A around -inf 24.1%

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

   if 1.00000000000000007e-68 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000012e63

   1. Initial program 40.5%

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

   3. Taylor expanded in F around 0 45.6%

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

      1. mul-1-neg 45.6%

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

      2. *-commutative 45.6%

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

      3. distribute-rgt-neg-in 45.6%

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

      4. associate-/l* 45.7%

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

      5. cancel-sign-sub-inv 45.7%

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

      6. metadata-eval 45.7%

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

      7. +-commutative 45.7%

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

   5. Simplified 60.6%

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

      1. pow1 60.6%

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

   7. Applied egg-rr 59.6%

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

      1. unpow1 59.6%

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

      2. unpow1/2 59.6%

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

      3. associate-+l+ 60.5%

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

   9. Simplified 60.5%

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

   if 2.00000000000000012e63 < (pow.f64 B #s(literal 2 binary64)) < 5e128

   1. Initial program 21.9%

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

   3. Taylor expanded in F around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. *-commutative 21.8%

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

      3. distribute-rgt-neg-in 21.8%

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

      4. associate-/l* 28.0%

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

      5. cancel-sign-sub-inv 28.0%

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

      6. metadata-eval 28.0%

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

      7. +-commutative 28.0%

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

   5. Simplified 35.7%

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

   6. Taylor expanded in A around -inf 47.5%

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

   if 5e128 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e305

   1. Initial program 16.7%

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

   3. Simplified 31.1%

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

      1. associate-*r* 31.1%

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

      2. associate-+r+ 31.0%

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

      3. hypot-undefine 16.7%

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

      4. unpow2 16.7%

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

      5. unpow2 16.7%

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

      6. +-commutative 16.7%

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

      7. sqrt-prod 30.8%

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

      8. *-commutative 30.8%

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

      9. associate-*r* 30.8%

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

      10. associate-+l+ 30.8%

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

   6. Applied egg-rr 50.5%

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

      1. clear-num 50.6%

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

      2. inv-pow 50.6%

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

      3. sqrt-unprod 31.1%

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

      4. associate-*l* 31.1%

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

      5. associate-*r* 31.1%

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

      6. associate-+r+ 31.0%

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

   8. Applied egg-rr 31.0%

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

      1. unpow-1 31.0%

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

      2. associate-*l* 11.5%

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

   10. Simplified 11.5%

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

   11. Taylor expanded in A around 0 25.6%

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

       1. mul-1-neg 25.6%

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

       2. unpow2 25.6%

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

       3. unpow2 25.6%

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

       4. hypot-define 31.3%

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

   13. Simplified 31.3%

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

   if 1.9999999999999999e305 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf 29.0%

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

      1. mul-1-neg 29.0%

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

      2. distribute-rgt-neg-in 29.0%

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

   5. Simplified 29.0%

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

      1. distribute-rgt-neg-out 29.0%

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

      2. pow1/2 29.0%

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

      3. pow1/2 29.0%

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

      4. pow-prod-down 29.2%

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

   7. Applied egg-rr 29.2%

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

      1. unpow1/2 29.2%

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

      2. associate-*l/ 29.2%

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

      3. sqrt-div 42.2%

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

   9. Applied egg-rr 42.2%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-68}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{\mathsf{fma}\left(A \cdot C, -4, {B}^{2}\right)}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{1}{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_3 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 7.5 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(2 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 7.5 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot t\_3\\ \mathbf{elif}\;B\_m \leq 2.35 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 6.5 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{{B\_m}^{2} + A \cdot \left(C \cdot -4\right)}} \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_3 (- (sqrt 2.0))))
   (if (<= B_m 7.5e-204)
     (/ (sqrt (* t_2 (* 2.0 C))) t_1)
     (if (<= B_m 7.5e-147)
       (* (sqrt (* F (/ -0.5 A))) t_3)
       (if (<= B_m 2.35e-29)
         (/ (sqrt (* t_2 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C)))) t_1)
         (if (<= B_m 6.5e+138)
           (*
            (sqrt
             (*
              F
              (/
               (+ A (+ C (hypot B_m (- A C))))
               (+ (pow B_m 2.0) (* A (* C -4.0))))))
            t_3)
           (/ (sqrt (* 2.0 F)) (- (sqrt 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 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_3 = -sqrt(2.0);
	double tmp;
	if (B_m <= 7.5e-204) {
		tmp = sqrt((t_2 * (2.0 * C))) / t_1;
	} else if (B_m <= 7.5e-147) {
		tmp = sqrt((F * (-0.5 / A))) * t_3;
	} else if (B_m <= 2.35e-29) {
		tmp = sqrt((t_2 * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_1;
	} else if (B_m <= 6.5e+138) {
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / (pow(B_m, 2.0) + (A * (C * -4.0)))))) * t_3;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

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 t_0 = (4.0 * A) * C;
	double t_1 = t_0 - Math.pow(B_m, 2.0);
	double t_2 = 2.0 * ((Math.pow(B_m, 2.0) - t_0) * F);
	double t_3 = -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 7.5e-204) {
		tmp = Math.sqrt((t_2 * (2.0 * C))) / t_1;
	} else if (B_m <= 7.5e-147) {
		tmp = Math.sqrt((F * (-0.5 / A))) * t_3;
	} else if (B_m <= 2.35e-29) {
		tmp = Math.sqrt((t_2 * ((-0.5 * (Math.pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_1;
	} else if (B_m <= 6.5e+138) {
		tmp = Math.sqrt((F * ((A + (C + Math.hypot(B_m, (A - C)))) / (Math.pow(B_m, 2.0) + (A * (C * -4.0)))))) * t_3;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(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):
	t_0 = (4.0 * A) * C
	t_1 = t_0 - math.pow(B_m, 2.0)
	t_2 = 2.0 * ((math.pow(B_m, 2.0) - t_0) * F)
	t_3 = -math.sqrt(2.0)
	tmp = 0
	if B_m <= 7.5e-204:
		tmp = math.sqrt((t_2 * (2.0 * C))) / t_1
	elif B_m <= 7.5e-147:
		tmp = math.sqrt((F * (-0.5 / A))) * t_3
	elif B_m <= 2.35e-29:
		tmp = math.sqrt((t_2 * ((-0.5 * (math.pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_1
	elif B_m <= 6.5e+138:
		tmp = math.sqrt((F * ((A + (C + math.hypot(B_m, (A - C)))) / (math.pow(B_m, 2.0) + (A * (C * -4.0)))))) * t_3
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_3 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 7.5e-204)
		tmp = Float64(sqrt(Float64(t_2 * Float64(2.0 * C))) / t_1);
	elseif (B_m <= 7.5e-147)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * t_3);
	elseif (B_m <= 2.35e-29)
		tmp = Float64(sqrt(Float64(t_2 * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / t_1);
	elseif (B_m <= 6.5e+138)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / Float64((B_m ^ 2.0) + Float64(A * Float64(C * -4.0)))))) * t_3);
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(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)
	t_0 = (4.0 * A) * C;
	t_1 = t_0 - (B_m ^ 2.0);
	t_2 = 2.0 * (((B_m ^ 2.0) - t_0) * F);
	t_3 = -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 7.5e-204)
		tmp = sqrt((t_2 * (2.0 * C))) / t_1;
	elseif (B_m <= 7.5e-147)
		tmp = sqrt((F * (-0.5 / A))) * t_3;
	elseif (B_m <= 2.35e-29)
		tmp = sqrt((t_2 * ((-0.5 * ((B_m ^ 2.0) / A)) + (2.0 * C)))) / t_1;
	elseif (B_m <= 6.5e+138)
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / ((B_m ^ 2.0) + (A * (C * -4.0)))))) * t_3;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 7.5e-204], N[(N[Sqrt[N[(t$95$2 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 7.5e-147], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[B$95$m, 2.35e-29], N[(N[Sqrt[N[(t$95$2 * 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$1), $MachinePrecision], If[LessEqual[B$95$m, 6.5e+138], N[(N[Sqrt[N[(F * N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < 7.5000000000000003e-204

   1. Initial program 18.6%

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

   3. Taylor expanded in A around -inf 14.9%

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

   if 7.5000000000000003e-204 < B < 7.50000000000000047e-147

   1. Initial program 18.9%

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

   3. Taylor expanded in F around 0 18.0%

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

      1. mul-1-neg 18.0%

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

      2. *-commutative 18.0%

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

      3. distribute-rgt-neg-in 18.0%

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

      4. associate-/l* 17.8%

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

      5. cancel-sign-sub-inv 17.8%

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

      6. metadata-eval 17.8%

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

      7. +-commutative 17.8%

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

   5. Simplified 35.0%

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

   6. Taylor expanded in A around -inf 35.2%

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

   if 7.50000000000000047e-147 < B < 2.3499999999999999e-29

   1. Initial program 30.7%

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

   3. Taylor expanded in A around -inf 35.4%

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

   if 2.3499999999999999e-29 < B < 6.50000000000000054e138

   1. Initial program 29.2%

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

   3. Taylor expanded in F around 0 33.4%

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

      1. mul-1-neg 33.4%

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

      2. *-commutative 33.4%

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

      3. distribute-rgt-neg-in 33.4%

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

      4. associate-/l* 37.7%

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

      5. cancel-sign-sub-inv 37.7%

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

      6. metadata-eval 37.7%

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

      7. +-commutative 37.7%

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

   5. Simplified 52.0%

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

      1. fma-undefine 52.0%

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

      2. *-commutative 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-*r* 52.0%

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

   7. Applied egg-rr 52.0%

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

   if 6.50000000000000054e138 < B

   1. Initial program 0.2%

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

   3. Taylor expanded in B around inf 48.7%

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

      1. mul-1-neg 48.7%

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

      2. distribute-rgt-neg-in 48.7%

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

   5. Simplified 48.7%

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

      1. distribute-rgt-neg-out 48.7%

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

      2. pow1/2 48.7%

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

      3. pow1/2 48.7%

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

      4. pow-prod-down 48.9%

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

   7. Applied egg-rr 48.9%

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

      1. unpow1/2 48.9%

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

      2. associate-*l/ 48.9%

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

      3. sqrt-div 71.2%

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

   9. Applied egg-rr 71.2%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.5 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{{B}^{2} + A \cdot \left(C \cdot -4\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ t_2 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 9 \cdot 10^{-204}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 7.5 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot t\_2\\ \mathbf{elif}\;B\_m \leq 5.6 \cdot 10^{-34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 5.5 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{{B\_m}^{2} + A \cdot \left(C \cdot -4\right)}} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
          (- t_0 (pow B_m 2.0))))
        (t_2 (- (sqrt 2.0))))
   (if (<= B_m 9e-204)
     t_1
     (if (<= B_m 7.5e-147)
       (* (sqrt (* F (/ -0.5 A))) t_2)
       (if (<= B_m 5.6e-34)
         t_1
         (if (<= B_m 5.5e+138)
           (*
            (sqrt
             (*
              F
              (/
               (+ A (+ C (hypot B_m (- A C))))
               (+ (pow B_m 2.0) (* A (* C -4.0))))))
            t_2)
           (/ (sqrt (* 2.0 F)) (- (sqrt 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 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	double t_2 = -sqrt(2.0);
	double tmp;
	if (B_m <= 9e-204) {
		tmp = t_1;
	} else if (B_m <= 7.5e-147) {
		tmp = sqrt((F * (-0.5 / A))) * t_2;
	} else if (B_m <= 5.6e-34) {
		tmp = t_1;
	} else if (B_m <= 5.5e+138) {
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / (pow(B_m, 2.0) + (A * (C * -4.0)))))) * t_2;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

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 t_0 = (4.0 * A) * C;
	double t_1 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	double t_2 = -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 9e-204) {
		tmp = t_1;
	} else if (B_m <= 7.5e-147) {
		tmp = Math.sqrt((F * (-0.5 / A))) * t_2;
	} else if (B_m <= 5.6e-34) {
		tmp = t_1;
	} else if (B_m <= 5.5e+138) {
		tmp = Math.sqrt((F * ((A + (C + Math.hypot(B_m, (A - C)))) / (Math.pow(B_m, 2.0) + (A * (C * -4.0)))))) * t_2;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(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):
	t_0 = (4.0 * A) * C
	t_1 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	t_2 = -math.sqrt(2.0)
	tmp = 0
	if B_m <= 9e-204:
		tmp = t_1
	elif B_m <= 7.5e-147:
		tmp = math.sqrt((F * (-0.5 / A))) * t_2
	elif B_m <= 5.6e-34:
		tmp = t_1
	elif B_m <= 5.5e+138:
		tmp = math.sqrt((F * ((A + (C + math.hypot(B_m, (A - C)))) / (math.pow(B_m, 2.0) + (A * (C * -4.0)))))) * t_2
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)))
	t_2 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 9e-204)
		tmp = t_1;
	elseif (B_m <= 7.5e-147)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * t_2);
	elseif (B_m <= 5.6e-34)
		tmp = t_1;
	elseif (B_m <= 5.5e+138)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / Float64((B_m ^ 2.0) + Float64(A * Float64(C * -4.0)))))) * t_2);
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(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)
	t_0 = (4.0 * A) * C;
	t_1 = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	t_2 = -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 9e-204)
		tmp = t_1;
	elseif (B_m <= 7.5e-147)
		tmp = sqrt((F * (-0.5 / A))) * t_2;
	elseif (B_m <= 5.6e-34)
		tmp = t_1;
	elseif (B_m <= 5.5e+138)
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / ((B_m ^ 2.0) + (A * (C * -4.0)))))) * t_2;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 9e-204], t$95$1, If[LessEqual[B$95$m, 7.5e-147], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[B$95$m, 5.6e-34], t$95$1, If[LessEqual[B$95$m, 5.5e+138], N[(N[Sqrt[N[(F * N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 8.99999999999999948e-204 or 7.50000000000000047e-147 < B < 5.59999999999999994e-34

   1. Initial program 20.4%

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

   3. Taylor expanded in A around -inf 17.3%

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

   if 8.99999999999999948e-204 < B < 7.50000000000000047e-147

   1. Initial program 18.9%

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

   3. Taylor expanded in F around 0 18.0%

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

      1. mul-1-neg 18.0%

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

      2. *-commutative 18.0%

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

      3. distribute-rgt-neg-in 18.0%

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

      4. associate-/l* 17.8%

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

      5. cancel-sign-sub-inv 17.8%

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

      6. metadata-eval 17.8%

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

      7. +-commutative 17.8%

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

   5. Simplified 35.0%

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

   6. Taylor expanded in A around -inf 35.2%

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

   if 5.59999999999999994e-34 < B < 5.4999999999999999e138

   1. Initial program 29.2%

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

   3. Taylor expanded in F around 0 33.4%

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

      1. mul-1-neg 33.4%

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

      2. *-commutative 33.4%

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

      3. distribute-rgt-neg-in 33.4%

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

      4. associate-/l* 37.7%

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

      5. cancel-sign-sub-inv 37.7%

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

      6. metadata-eval 37.7%

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

      7. +-commutative 37.7%

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

   5. Simplified 52.0%

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

      1. fma-undefine 52.0%

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

      2. *-commutative 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-*r* 52.0%

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

   7. Applied egg-rr 52.0%

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

   if 5.4999999999999999e138 < B

   1. Initial program 0.2%

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

   3. Taylor expanded in B around inf 48.7%

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

      1. mul-1-neg 48.7%

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

      2. distribute-rgt-neg-in 48.7%

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

   5. Simplified 48.7%

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

      1. distribute-rgt-neg-out 48.7%

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

      2. pow1/2 48.7%

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

      3. pow1/2 48.7%

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

      4. pow-prod-down 48.9%

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

   7. Applied egg-rr 48.9%

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

      1. unpow1/2 48.9%

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

      2. associate-*l/ 48.9%

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

      3. sqrt-div 71.2%

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

   9. Applied egg-rr 71.2%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{{B}^{2} + A \cdot \left(C \cdot -4\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{if}\;B\_m \leq 4.5 \cdot 10^{-203}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 8.5 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 7.5 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 10^{+184}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{1}{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
          (- t_0 (pow B_m 2.0)))))
   (if (<= B_m 4.5e-203)
     t_1
     (if (<= B_m 8.5e-147)
       (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
       (if (<= B_m 7.5e-35)
         t_1
         (if (<= B_m 1e+184)
           (/
            -1.0
            (* (/ B_m (sqrt 2.0)) (sqrt (/ 1.0 (* F (+ C (hypot B_m C)))))))
           (/ (sqrt (* 2.0 F)) (- (sqrt 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 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	double tmp;
	if (B_m <= 4.5e-203) {
		tmp = t_1;
	} else if (B_m <= 8.5e-147) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else if (B_m <= 7.5e-35) {
		tmp = t_1;
	} else if (B_m <= 1e+184) {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt((1.0 / (F * (C + hypot(B_m, C))))));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

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 t_0 = (4.0 * A) * C;
	double t_1 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	double tmp;
	if (B_m <= 4.5e-203) {
		tmp = t_1;
	} else if (B_m <= 8.5e-147) {
		tmp = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	} else if (B_m <= 7.5e-35) {
		tmp = t_1;
	} else if (B_m <= 1e+184) {
		tmp = -1.0 / ((B_m / Math.sqrt(2.0)) * Math.sqrt((1.0 / (F * (C + Math.hypot(B_m, C))))));
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(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):
	t_0 = (4.0 * A) * C
	t_1 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	tmp = 0
	if B_m <= 4.5e-203:
		tmp = t_1
	elif B_m <= 8.5e-147:
		tmp = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	elif B_m <= 7.5e-35:
		tmp = t_1
	elif B_m <= 1e+184:
		tmp = -1.0 / ((B_m / math.sqrt(2.0)) * math.sqrt((1.0 / (F * (C + math.hypot(B_m, C))))))
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)))
	tmp = 0.0
	if (B_m <= 4.5e-203)
		tmp = t_1;
	elseif (B_m <= 8.5e-147)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 7.5e-35)
		tmp = t_1;
	elseif (B_m <= 1e+184)
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(1.0 / Float64(F * Float64(C + hypot(B_m, C)))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(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)
	t_0 = (4.0 * A) * C;
	t_1 = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	tmp = 0.0;
	if (B_m <= 4.5e-203)
		tmp = t_1;
	elseif (B_m <= 8.5e-147)
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	elseif (B_m <= 7.5e-35)
		tmp = t_1;
	elseif (B_m <= 1e+184)
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt((1.0 / (F * (C + hypot(B_m, C))))));
	else
		tmp = sqrt((2.0 * F)) / -sqrt(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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.5e-203], t$95$1, If[LessEqual[B$95$m, 8.5e-147], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 7.5e-35], t$95$1, If[LessEqual[B$95$m, 1e+184], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 4.5000000000000002e-203 or 8.5000000000000002e-147 < B < 7.5e-35

   1. Initial program 20.5%

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

   3. Taylor expanded in A around -inf 17.4%

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

   if 4.5000000000000002e-203 < B < 8.5000000000000002e-147

   1. Initial program 18.9%

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

   3. Taylor expanded in F around 0 18.0%

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

      1. mul-1-neg 18.0%

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

      2. *-commutative 18.0%

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

      3. distribute-rgt-neg-in 18.0%

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

      4. associate-/l* 17.8%

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

      5. cancel-sign-sub-inv 17.8%

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

      6. metadata-eval 17.8%

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

      7. +-commutative 17.8%

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

   5. Simplified 35.0%

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

   6. Taylor expanded in A around -inf 35.2%

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

   if 7.5e-35 < B < 1.00000000000000002e184

   1. Initial program 23.1%

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

   3. Simplified 28.6%

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

      1. associate-*r* 28.6%

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

      2. associate-+r+ 28.4%

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

      3. hypot-undefine 23.1%

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

      4. unpow2 23.1%

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

      5. unpow2 23.1%

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

      6. +-commutative 23.1%

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

      7. sqrt-prod 28.1%

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

      8. *-commutative 28.1%

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

      9. associate-*r* 28.1%

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

      10. associate-+l+ 28.1%

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

   6. Applied egg-rr 45.6%

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

      1. clear-num 45.5%

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

      2. inv-pow 45.5%

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

      3. sqrt-unprod 28.6%

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

      4. associate-*l* 28.6%

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

      5. associate-*r* 28.6%

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

      6. associate-+r+ 28.4%

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

   8. Applied egg-rr 28.4%

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

      1. unpow-1 28.4%

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

      2. associate-*l* 21.6%

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

   10. Simplified 21.6%

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

   11. Taylor expanded in A around 0 28.3%

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

       1. mul-1-neg 28.3%

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

       2. unpow2 28.3%

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

       3. unpow2 28.3%

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

       4. hypot-define 38.7%

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

   13. Simplified 38.7%

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

   if 1.00000000000000002e184 < 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 54.8%

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

      1. mul-1-neg 54.8%

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

      2. distribute-rgt-neg-in 54.8%

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

   5. Simplified 54.8%

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

      1. distribute-rgt-neg-out 54.8%

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

      2. pow1/2 54.8%

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

      3. pow1/2 54.8%

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

      4. pow-prod-down 55.1%

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

   7. Applied egg-rr 55.1%

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

      1. unpow1/2 55.1%

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

      2. associate-*l/ 55.1%

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

      3. sqrt-div 82.1%

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

   9. Applied egg-rr 82.1%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.5 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 10^{+184}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{1}{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)\\ t_1 := \sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{if}\;B\_m \leq 1.16 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 4.6 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{t\_0} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{elif}\;B\_m \leq 1.4 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 1.1 \cdot 10^{+185}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{1}{t\_0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* F (+ C (hypot B_m C))))
        (t_1 (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))))
   (if (<= B_m 1.16e-22)
     t_1
     (if (<= B_m 4.6e+31)
       (* (sqrt t_0) (/ (sqrt 2.0) (- B_m)))
       (if (<= B_m 1.4e+65)
         t_1
         (if (<= B_m 1.1e+185)
           (/ -1.0 (* (/ B_m (sqrt 2.0)) (sqrt (/ 1.0 t_0))))
           (/ (sqrt (* 2.0 F)) (- (sqrt 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 = F * (C + hypot(B_m, C));
	double t_1 = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	double tmp;
	if (B_m <= 1.16e-22) {
		tmp = t_1;
	} else if (B_m <= 4.6e+31) {
		tmp = sqrt(t_0) * (sqrt(2.0) / -B_m);
	} else if (B_m <= 1.4e+65) {
		tmp = t_1;
	} else if (B_m <= 1.1e+185) {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt((1.0 / t_0)));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

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 t_0 = F * (C + Math.hypot(B_m, C));
	double t_1 = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 1.16e-22) {
		tmp = t_1;
	} else if (B_m <= 4.6e+31) {
		tmp = Math.sqrt(t_0) * (Math.sqrt(2.0) / -B_m);
	} else if (B_m <= 1.4e+65) {
		tmp = t_1;
	} else if (B_m <= 1.1e+185) {
		tmp = -1.0 / ((B_m / Math.sqrt(2.0)) * Math.sqrt((1.0 / t_0)));
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(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):
	t_0 = F * (C + math.hypot(B_m, C))
	t_1 = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	tmp = 0
	if B_m <= 1.16e-22:
		tmp = t_1
	elif B_m <= 4.6e+31:
		tmp = math.sqrt(t_0) * (math.sqrt(2.0) / -B_m)
	elif B_m <= 1.4e+65:
		tmp = t_1
	elif B_m <= 1.1e+185:
		tmp = -1.0 / ((B_m / math.sqrt(2.0)) * math.sqrt((1.0 / t_0)))
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(F * Float64(C + hypot(B_m, C)))
	t_1 = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)))
	tmp = 0.0
	if (B_m <= 1.16e-22)
		tmp = t_1;
	elseif (B_m <= 4.6e+31)
		tmp = Float64(sqrt(t_0) * Float64(sqrt(2.0) / Float64(-B_m)));
	elseif (B_m <= 1.4e+65)
		tmp = t_1;
	elseif (B_m <= 1.1e+185)
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(1.0 / t_0))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(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)
	t_0 = F * (C + hypot(B_m, C));
	t_1 = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 1.16e-22)
		tmp = t_1;
	elseif (B_m <= 4.6e+31)
		tmp = sqrt(t_0) * (sqrt(2.0) / -B_m);
	elseif (B_m <= 1.4e+65)
		tmp = t_1;
	elseif (B_m <= 1.1e+185)
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt((1.0 / t_0)));
	else
		tmp = sqrt((2.0 * F)) / -sqrt(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_] := Block[{t$95$0 = N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[B$95$m, 1.16e-22], t$95$1, If[LessEqual[B$95$m, 4.6e+31], N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.4e+65], t$95$1, If[LessEqual[B$95$m, 1.1e+185], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.1600000000000001e-22 or 4.5999999999999999e31 < B < 1.3999999999999999e65

   1. Initial program 21.2%

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

   3. Taylor expanded in F around 0 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. distribute-rgt-neg-in 17.9%

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

      4. associate-/l* 18.9%

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

      5. cancel-sign-sub-inv 18.9%

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

      6. metadata-eval 18.9%

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

      7. +-commutative 18.9%

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

   5. Simplified 29.5%

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

   6. Taylor expanded in A around -inf 22.0%

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

   if 1.1600000000000001e-22 < B < 4.5999999999999999e31

   1. Initial program 28.8%

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

   3. Taylor expanded in A around 0 25.2%

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

      1. mul-1-neg 25.2%

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

      2. *-commutative 25.2%

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

      3. distribute-rgt-neg-in 25.2%

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

      4. unpow2 25.2%

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

      5. unpow2 25.2%

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

      6. hypot-define 25.6%

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

   5. Simplified 25.6%

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

   if 1.3999999999999999e65 < B < 1.1e185

   1. Initial program 16.3%

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

   3. Simplified 26.9%

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

      1. associate-*r* 26.9%

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

      2. associate-+r+ 26.8%

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

      3. hypot-undefine 16.3%

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

      4. unpow2 16.3%

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

      5. unpow2 16.3%

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

      6. +-commutative 16.3%

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

      7. sqrt-prod 26.8%

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

      8. *-commutative 26.8%

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

      9. associate-*r* 26.8%

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

      10. associate-+l+ 26.8%

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

   6. Applied egg-rr 41.0%

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

      1. clear-num 41.0%

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

      2. inv-pow 41.0%

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

      3. sqrt-unprod 26.8%

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

      4. associate-*l* 26.8%

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

      5. associate-*r* 26.8%

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

      6. associate-+r+ 26.7%

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

   8. Applied egg-rr 26.7%

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

      1. unpow-1 26.7%

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

      2. associate-*l* 12.9%

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

   10. Simplified 12.9%

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

   11. Taylor expanded in A around 0 31.8%

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

       1. mul-1-neg 31.8%

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

       2. unpow2 31.8%

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

       3. unpow2 31.8%

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

       4. hypot-define 53.4%

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

   13. Simplified 53.4%

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

   if 1.1e185 < 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 54.8%

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

      1. mul-1-neg 54.8%

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

      2. distribute-rgt-neg-in 54.8%

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

   5. Simplified 54.8%

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

      1. distribute-rgt-neg-out 54.8%

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

      2. pow1/2 54.8%

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

      3. pow1/2 54.8%

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

      4. pow-prod-down 55.1%

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

   7. Applied egg-rr 55.1%

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

      1. unpow1/2 55.1%

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

      2. associate-*l/ 55.1%

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

      3. sqrt-div 82.1%

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

   9. Applied egg-rr 82.1%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.16 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{+185}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{1}{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)\\ \mathbf{if}\;B\_m \leq 1.16 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{F \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot t\_0\right)\right)}}}\\ \mathbf{elif}\;B\_m \leq 5.1 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{t\_1} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{elif}\;B\_m \leq 4.1 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 3.2 \cdot 10^{+184}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{1}{t\_1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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)))) (t_1 (* F (+ C (hypot B_m C)))))
   (if (<= B_m 1.16e-29)
     (/ -1.0 (/ t_0 (sqrt (* F (* (* 2.0 C) (* 2.0 t_0))))))
     (if (<= B_m 5.1e+31)
       (* (sqrt t_1) (/ (sqrt 2.0) (- B_m)))
       (if (<= B_m 4.1e+64)
         (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
         (if (<= B_m 3.2e+184)
           (/ -1.0 (* (/ B_m (sqrt 2.0)) (sqrt (/ 1.0 t_1))))
           (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = F * (C + hypot(B_m, C));
	double tmp;
	if (B_m <= 1.16e-29) {
		tmp = -1.0 / (t_0 / sqrt((F * ((2.0 * C) * (2.0 * t_0)))));
	} else if (B_m <= 5.1e+31) {
		tmp = sqrt(t_1) * (sqrt(2.0) / -B_m);
	} else if (B_m <= 4.1e+64) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else if (B_m <= 3.2e+184) {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt((1.0 / t_1)));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(F * Float64(C + hypot(B_m, C)))
	tmp = 0.0
	if (B_m <= 1.16e-29)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(F * Float64(Float64(2.0 * C) * Float64(2.0 * t_0))))));
	elseif (B_m <= 5.1e+31)
		tmp = Float64(sqrt(t_1) * Float64(sqrt(2.0) / Float64(-B_m)));
	elseif (B_m <= 4.1e+64)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 3.2e+184)
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(1.0 / t_1))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.16e-29], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(F * N[(N[(2.0 * C), $MachinePrecision] * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 5.1e+31], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.1e+64], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 3.2e+184], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < 1.15999999999999996e-29

   1. Initial program 20.4%

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

   3. Simplified 28.2%

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

      1. associate-*r* 28.2%

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

      2. associate-+r+ 27.4%

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

      3. hypot-undefine 20.4%

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

      4. unpow2 20.4%

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

      5. unpow2 20.4%

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

      6. +-commutative 20.4%

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

      7. sqrt-prod 22.8%

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

      8. *-commutative 22.8%

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

      9. associate-*r* 22.8%

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

      10. associate-+l+ 23.0%

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

   6. Applied egg-rr 33.7%

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

      1. clear-num 33.7%

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

      2. inv-pow 33.7%

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

      3. sqrt-unprod 28.2%

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

      4. associate-*l* 28.2%

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

      5. associate-*r* 28.2%

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

      6. associate-+r+ 27.4%

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

   8. Applied egg-rr 27.4%

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

      1. unpow-1 27.4%

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

      2. associate-*l* 24.1%

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

   10. Simplified 24.1%

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

   11. Taylor expanded in A around -inf 15.2%

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

   if 1.15999999999999996e-29 < B < 5.0999999999999997e31

   1. Initial program 31.3%

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

   3. Taylor expanded in A around 0 28.0%

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

      1. mul-1-neg 28.0%

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

      2. *-commutative 28.0%

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

      3. distribute-rgt-neg-in 28.0%

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

      4. unpow2 28.0%

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

      5. unpow2 28.0%

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

      6. hypot-define 28.4%

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

   5. Simplified 28.4%

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

   if 5.0999999999999997e31 < B < 4.09999999999999978e64

   1. Initial program 29.3%

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

   3. Taylor expanded in F around 0 29.1%

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

      1. mul-1-neg 29.1%

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

      2. *-commutative 29.1%

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

      3. distribute-rgt-neg-in 29.1%

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

      4. associate-/l* 37.6%

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

      5. cancel-sign-sub-inv 37.6%

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

      6. metadata-eval 37.6%

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

      7. +-commutative 37.6%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in A around -inf 46.5%

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

   if 4.09999999999999978e64 < B < 3.19999999999999983e184

   1. Initial program 16.3%

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

   3. Simplified 26.9%

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

      1. associate-*r* 26.9%

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

      2. associate-+r+ 26.8%

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

      3. hypot-undefine 16.3%

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

      4. unpow2 16.3%

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

      5. unpow2 16.3%

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

      6. +-commutative 16.3%

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

      7. sqrt-prod 26.8%

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

      8. *-commutative 26.8%

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

      9. associate-*r* 26.8%

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

      10. associate-+l+ 26.8%

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

   6. Applied egg-rr 41.0%

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

      1. clear-num 41.0%

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

      2. inv-pow 41.0%

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

      3. sqrt-unprod 26.8%

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

      4. associate-*l* 26.8%

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

      5. associate-*r* 26.8%

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

      6. associate-+r+ 26.7%

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

   8. Applied egg-rr 26.7%

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

      1. unpow-1 26.7%

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

      2. associate-*l* 12.9%

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

   10. Simplified 12.9%

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

   11. Taylor expanded in A around 0 31.8%

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

       1. mul-1-neg 31.8%

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

       2. unpow2 31.8%

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

       3. unpow2 31.8%

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

       4. hypot-define 53.4%

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

   13. Simplified 53.4%

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

   if 3.19999999999999983e184 < 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 54.8%

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

      1. mul-1-neg 54.8%

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

      2. distribute-rgt-neg-in 54.8%

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

   5. Simplified 54.8%

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

      1. distribute-rgt-neg-out 54.8%

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

      2. pow1/2 54.8%

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

      3. pow1/2 54.8%

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

      4. pow-prod-down 55.1%

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

   7. Applied egg-rr 55.1%

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

      1. unpow1/2 55.1%

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

      2. associate-*l/ 55.1%

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

      3. sqrt-div 82.1%

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

   9. Applied egg-rr 82.1%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.16 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{F \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)}}}\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{+184}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{1}{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ t_1 := \sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{if}\;B\_m \leq 5.5 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 6 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;B\_m \leq 1.1 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 10^{+184}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* (sqrt (* F (+ C (hypot B_m C)))) (/ (sqrt 2.0) (- B_m))))
        (t_1 (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))))
   (if (<= B_m 5.5e-22)
     t_1
     (if (<= B_m 6e+31)
       t_0
       (if (<= B_m 1.1e+73)
         t_1
         (if (<= B_m 1e+184) t_0 (/ (sqrt (* 2.0 F)) (- (sqrt 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 = sqrt((F * (C + hypot(B_m, C)))) * (sqrt(2.0) / -B_m);
	double t_1 = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	double tmp;
	if (B_m <= 5.5e-22) {
		tmp = t_1;
	} else if (B_m <= 6e+31) {
		tmp = t_0;
	} else if (B_m <= 1.1e+73) {
		tmp = t_1;
	} else if (B_m <= 1e+184) {
		tmp = t_0;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

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 t_0 = Math.sqrt((F * (C + Math.hypot(B_m, C)))) * (Math.sqrt(2.0) / -B_m);
	double t_1 = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 5.5e-22) {
		tmp = t_1;
	} else if (B_m <= 6e+31) {
		tmp = t_0;
	} else if (B_m <= 1.1e+73) {
		tmp = t_1;
	} else if (B_m <= 1e+184) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(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):
	t_0 = math.sqrt((F * (C + math.hypot(B_m, C)))) * (math.sqrt(2.0) / -B_m)
	t_1 = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	tmp = 0
	if B_m <= 5.5e-22:
		tmp = t_1
	elif B_m <= 6e+31:
		tmp = t_0
	elif B_m <= 1.1e+73:
		tmp = t_1
	elif B_m <= 1e+184:
		tmp = t_0
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(Float64(F * Float64(C + hypot(B_m, C)))) * Float64(sqrt(2.0) / Float64(-B_m)))
	t_1 = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)))
	tmp = 0.0
	if (B_m <= 5.5e-22)
		tmp = t_1;
	elseif (B_m <= 6e+31)
		tmp = t_0;
	elseif (B_m <= 1.1e+73)
		tmp = t_1;
	elseif (B_m <= 1e+184)
		tmp = t_0;
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(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)
	t_0 = sqrt((F * (C + hypot(B_m, C)))) * (sqrt(2.0) / -B_m);
	t_1 = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 5.5e-22)
		tmp = t_1;
	elseif (B_m <= 6e+31)
		tmp = t_0;
	elseif (B_m <= 1.1e+73)
		tmp = t_1;
	elseif (B_m <= 1e+184)
		tmp = t_0;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(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_] := Block[{t$95$0 = N[(N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[B$95$m, 5.5e-22], t$95$1, If[LessEqual[B$95$m, 6e+31], t$95$0, If[LessEqual[B$95$m, 1.1e+73], t$95$1, If[LessEqual[B$95$m, 1e+184], t$95$0, N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 5.5000000000000001e-22 or 5.99999999999999978e31 < B < 1.1e73

   1. Initial program 21.2%

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

   3. Taylor expanded in F around 0 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. distribute-rgt-neg-in 17.9%

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

      4. associate-/l* 18.9%

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

      5. cancel-sign-sub-inv 18.9%

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

      6. metadata-eval 18.9%

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

      7. +-commutative 18.9%

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

   5. Simplified 29.5%

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

   6. Taylor expanded in A around -inf 22.0%

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

   if 5.5000000000000001e-22 < B < 5.99999999999999978e31 or 1.1e73 < B < 1.00000000000000002e184

   1. Initial program 20.7%

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

   3. Taylor expanded in A around 0 29.3%

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

      1. mul-1-neg 29.3%

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

      2. *-commutative 29.3%

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

      3. distribute-rgt-neg-in 29.3%

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

      4. unpow2 29.3%

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

      5. unpow2 29.3%

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

      6. hypot-define 43.4%

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

   5. Simplified 43.4%

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

   if 1.00000000000000002e184 < 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 54.8%

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

      1. mul-1-neg 54.8%

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

      2. distribute-rgt-neg-in 54.8%

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

   5. Simplified 54.8%

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

      1. distribute-rgt-neg-out 54.8%

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

      2. pow1/2 54.8%

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

      3. pow1/2 54.8%

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

      4. pow-prod-down 55.1%

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

   7. Applied egg-rr 55.1%

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

      1. unpow1/2 55.1%

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

      2. associate-*l/ 55.1%

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

      3. sqrt-div 82.1%

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

   9. Applied egg-rr 82.1%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.5 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 6 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 10^{+184}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{if}\;B\_m \leq 1.3 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;B\_m \leq 940000000000:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(B\_m + C\right)}\\ \mathbf{elif}\;B\_m \leq 1.75 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))))
   (if (<= B_m 1.3e-21)
     t_0
     (if (<= B_m 940000000000.0)
       (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (+ B_m C))))
       (if (<= B_m 1.75e+79) t_0 (/ (sqrt (* 2.0 F)) (- (sqrt 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 = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	double tmp;
	if (B_m <= 1.3e-21) {
		tmp = t_0;
	} else if (B_m <= 940000000000.0) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (B_m + C)));
	} else if (B_m <= 1.75e+79) {
		tmp = t_0;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((f * ((-0.5d0) / a))) * -sqrt(2.0d0)
    if (b_m <= 1.3d-21) then
        tmp = t_0
    else if (b_m <= 940000000000.0d0) then
        tmp = (sqrt(2.0d0) / -b_m) * sqrt((f * (b_m + c)))
    else if (b_m <= 1.75d+79) then
        tmp = t_0
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(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 t_0 = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 1.3e-21) {
		tmp = t_0;
	} else if (B_m <= 940000000000.0) {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * (B_m + C)));
	} else if (B_m <= 1.75e+79) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(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):
	t_0 = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	tmp = 0
	if B_m <= 1.3e-21:
		tmp = t_0
	elif B_m <= 940000000000.0:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * (B_m + C)))
	elif B_m <= 1.75e+79:
		tmp = t_0
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)))
	tmp = 0.0
	if (B_m <= 1.3e-21)
		tmp = t_0;
	elseif (B_m <= 940000000000.0)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(B_m + C))));
	elseif (B_m <= 1.75e+79)
		tmp = t_0;
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(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)
	t_0 = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 1.3e-21)
		tmp = t_0;
	elseif (B_m <= 940000000000.0)
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (B_m + C)));
	elseif (B_m <= 1.75e+79)
		tmp = t_0;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(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_] := Block[{t$95$0 = N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[B$95$m, 1.3e-21], t$95$0, If[LessEqual[B$95$m, 940000000000.0], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(B$95$m + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.75e+79], t$95$0, N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.30000000000000009e-21 or 9.4e11 < B < 1.7499999999999999e79

   1. Initial program 21.7%

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

   3. Taylor expanded in F around 0 18.4%

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

      1. mul-1-neg 18.4%

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

      2. *-commutative 18.4%

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

      3. distribute-rgt-neg-in 18.4%

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

      4. associate-/l* 19.4%

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

      5. cancel-sign-sub-inv 19.4%

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

      6. metadata-eval 19.4%

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

      7. +-commutative 19.4%

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

   5. Simplified 30.7%

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

   6. Taylor expanded in A around -inf 21.9%

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

   if 1.30000000000000009e-21 < B < 9.4e11

   1. Initial program 22.3%

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

   3. Taylor expanded in B around inf 22.4%

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

   4. Taylor expanded in A around 0 22.4%

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

      1. mul-1-neg 22.4%

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

   6. Simplified 22.4%

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

   if 1.7499999999999999e79 < B

   1. Initial program 7.6%

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

   3. Taylor expanded in B around inf 48.9%

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

      1. mul-1-neg 48.9%

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

      2. distribute-rgt-neg-in 48.9%

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

   5. Simplified 48.9%

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

      1. distribute-rgt-neg-out 48.9%

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

      2. pow1/2 48.9%

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

      3. pow1/2 48.9%

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

      4. pow-prod-down 49.1%

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

   7. Applied egg-rr 49.1%

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

      1. unpow1/2 49.1%

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

      2. associate-*l/ 49.1%

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

      3. sqrt-div 67.2%

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

   9. Applied egg-rr 67.2%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.3 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 940000000000:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.46 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 1.46e+78)
   (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
   (/ (sqrt (* 2.0 F)) (- (sqrt 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 <= 1.46e+78) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(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 <= 1.46d+78) then
        tmp = sqrt((f * ((-0.5d0) / a))) * -sqrt(2.0d0)
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(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 <= 1.46e+78) {
		tmp = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(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 <= 1.46e+78:
		tmp = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(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 <= 1.46e+78)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(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 <= 1.46e+78)
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	else
		tmp = sqrt((2.0 * F)) / -sqrt(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, 1.46e+78], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.46000000000000005e78

   1. Initial program 21.7%

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

   3. Taylor expanded in F around 0 18.6%

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

      1. mul-1-neg 18.6%

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

      2. *-commutative 18.6%

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

      3. distribute-rgt-neg-in 18.6%

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

      4. associate-/l* 19.5%

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

      5. cancel-sign-sub-inv 19.5%

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

      6. metadata-eval 19.5%

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

      7. +-commutative 19.5%

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

   5. Simplified 30.9%

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

   6. Taylor expanded in A around -inf 21.4%

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

   if 1.46000000000000005e78 < B

   1. Initial program 7.6%

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

   3. Taylor expanded in B around inf 48.9%

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

      1. mul-1-neg 48.9%

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

      2. distribute-rgt-neg-in 48.9%

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

   5. Simplified 48.9%

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

      1. distribute-rgt-neg-out 48.9%

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

      2. pow1/2 48.9%

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

      3. pow1/2 48.9%

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

      4. pow-prod-down 49.1%

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

   7. Applied egg-rr 49.1%

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

      1. unpow1/2 49.1%

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

      2. associate-*l/ 49.1%

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

      3. sqrt-div 67.2%

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

   9. Applied egg-rr 67.2%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.46 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in B around inf 15.4%

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

   1. mul-1-neg 15.4%

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

   2. distribute-rgt-neg-in 15.4%

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

5. Simplified 15.4%

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

   1. distribute-rgt-neg-out 15.4%

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

   2. pow1/2 15.6%

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

   3. pow1/2 15.6%

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

   4. pow-prod-down 15.7%

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

7. Applied egg-rr 15.7%

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

   1. unpow1/2 15.4%

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

   2. associate-*l/ 15.4%

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

   3. sqrt-div 19.8%

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

9. Applied egg-rr 19.8%

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

10. Final simplification 19.8%

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

Alternative 15: 26.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -{\left(2 \cdot \frac{F}{B\_m}\right)}^{0.5} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (pow (* 2.0 (/ F B_m)) 0.5)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -pow((2.0 * (F / B_m)), 0.5);
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -((2.0d0 * (f / b_m)) ** 0.5d0)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.pow((2.0 * (F / B_m)), 0.5);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.pow((2.0 * (F / B_m)), 0.5)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-(Float64(2.0 * Float64(F / B_m)) ^ 0.5))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -((2.0 * (F / B_m)) ^ 0.5);
end

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in B around inf 15.4%

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

   1. mul-1-neg 15.4%

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

   2. distribute-rgt-neg-in 15.4%

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

5. Simplified 15.4%

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

   1. distribute-rgt-neg-out 15.4%

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

   2. pow1/2 15.6%

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

   3. pow1/2 15.6%

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

   4. pow-prod-down 15.7%

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

7. Applied egg-rr 15.7%

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

8. Final simplification 15.7%

   \[\leadsto -{\left(2 \cdot \frac{F}{B}\right)}^{0.5} \]
9. Add Preprocessing

Alternative 16: 26.8% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in B around inf 15.4%

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

   1. mul-1-neg 15.4%

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

   2. distribute-rgt-neg-in 15.4%

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

5. Simplified 15.4%

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

   1. distribute-rgt-neg-out 15.4%

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

   2. pow1/2 15.6%

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

   3. pow1/2 15.6%

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

   4. pow-prod-down 15.7%

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

7. Applied egg-rr 15.7%

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

   1. *-un-lft-identity 15.7%

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

   2. unpow1/2 15.4%

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

   3. associate-*l/ 15.4%

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

9. Applied egg-rr 15.4%

   \[\leadsto -\color{blue}{1 \cdot \sqrt{\frac{F \cdot 2}{B}}} \]
10. Step-by-step derivation

    1. *-lft-identity 15.4%

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

    2. associate-/l* 15.4%

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

11. Simplified 15.4%

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

12. Final simplification 15.4%

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

Alternative 17: 26.8% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in B around inf 15.4%

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

   1. mul-1-neg 15.4%

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

   2. distribute-rgt-neg-in 15.4%

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

5. Simplified 15.4%

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

   1. pow1 15.4%

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

   2. distribute-rgt-neg-out 15.4%

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

   3. pow1/2 15.6%

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

   4. pow1/2 15.6%

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

   5. pow-prod-down 15.7%

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

7. Applied egg-rr 15.7%

   \[\leadsto \color{blue}{{\left(-{\left(\frac{F}{B} \cdot 2\right)}^{0.5}\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 15.7%

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

   2. unpow1/2 15.4%

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

9. Simplified 15.4%

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

10. Final simplification 15.4%

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

Reproduce

herbie shell --seed 2024109 
(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))))