ABCF->ab-angle b

Percentage Accurate: 19.0% → 49.2%

Time: 34.8s

Alternatives: 17

Speedup: 5.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 49.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{F \cdot \left(\frac{1}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)} \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{{B\_m}^{2} \cdot -0.5}{C}\right)\right)\right)}}{-t\_0}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\_m\right)\right) \cdot \left(2 \cdot t\_3\right)} \cdot \sqrt{F}}{-t\_3}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \left(-\frac{F}{B\_m}\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0))))
        (t_3 (fma C (* A -4.0) (pow B_m 2.0))))
   (if (<= t_2 -1e-144)
     (*
      (sqrt
       (*
        F
        (*
         (/ 1.0 (fma -4.0 (* A C) (pow B_m 2.0)))
         (+ A (- C (hypot B_m (- A C)))))))
      (- (sqrt 2.0)))
     (if (<= t_2 5e-112)
       (/
        (sqrt (* (* F t_0) (* 2.0 (+ A (+ A (/ (* (pow B_m 2.0) -0.5) C))))))
        (- t_0))
       (if (<= t_2 INFINITY)
         (/
          (* (sqrt (* (- (+ A C) (hypot (- A C) B_m)) (* 2.0 t_3))) (sqrt F))
          (- t_3))
         (- (sqrt (* 2.0 (- (/ F B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double t_3 = fma(C, (A * -4.0), pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -1e-144) {
		tmp = sqrt((F * ((1.0 / fma(-4.0, (A * C), pow(B_m, 2.0))) * (A + (C - hypot(B_m, (A - C))))))) * -sqrt(2.0);
	} else if (t_2 <= 5e-112) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + (A + ((pow(B_m, 2.0) * -0.5) / C)))))) / -t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (sqrt((((A + C) - hypot((A - C), B_m)) * (2.0 * t_3))) * sqrt(F)) / -t_3;
	} else {
		tmp = -sqrt((2.0 * -(F / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = fma(C, Float64(A * -4.0), (B_m ^ 2.0))
	tmp = 0.0
	if (t_2 <= -1e-144)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(1.0 / fma(-4.0, Float64(A * C), (B_m ^ 2.0))) * Float64(A + Float64(C - hypot(B_m, Float64(A - C))))))) * Float64(-sqrt(2.0)));
	elseif (t_2 <= 5e-112)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(A + Float64(Float64((B_m ^ 2.0) * -0.5) / C)))))) / Float64(-t_0));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(A + C) - hypot(Float64(A - C), B_m)) * Float64(2.0 * t_3))) * sqrt(F)) / Float64(-t_3));
	else
		tmp = Float64(-sqrt(Float64(2.0 * Float64(-Float64(F / B_m)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-144], N[(N[Sqrt[N[(F * N[(N[(1.0 / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, 5e-112], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / (-t$95$3)), $MachinePrecision], (-N[Sqrt[N[(2.0 * (-N[(F / B$95$m), $MachinePrecision])), $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))) < -9.9999999999999995e-145

   1. Initial program 41.3%

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

   3. Taylor expanded in F around 0 46.6%

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

   5. Simplified 67.4%

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

      1. div-inv 67.4%

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

      2. associate--r- 68.0%

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

   7. Applied egg-rr 68.0%

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

   if -9.9999999999999995e-145 < (/.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))) < 5.00000000000000044e-112

   1. Initial program 11.8%

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

   3. Simplified 15.1%

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

   5. Taylor expanded in C around inf 26.6%

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

      1. associate-*r/ 26.6%

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

      2. mul-1-neg 26.6%

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

   7. Simplified 26.6%

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

   if 5.00000000000000044e-112 < (/.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 32.4%

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

   3. Simplified 45.0%

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

      1. pow1/2 45.0%

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

      2. *-commutative 45.0%

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

      3. unpow-prod-down 61.8%

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

   6. Applied egg-rr 61.8%

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

   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 F around 0 0.0%

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

   5. Simplified 4.0%

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

   6. Taylor expanded in B around inf 20.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-1}{B}}} \]
   7. Step-by-step derivation

      1. pow1 20.2%

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

      2. sqrt-unprod 20.2%

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

   8. Applied egg-rr 20.2%

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

      1. unpow1 20.2%

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

      2. associate-*r/ 20.3%

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

      3. *-commutative 20.3%

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

      4. mul-1-neg 20.3%

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

   10. Simplified 20.3%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{F \cdot \left(\frac{1}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)} \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{{B}^{2} \cdot -0.5}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)\right)} \cdot \sqrt{F}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \left(-\frac{F}{B}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 47.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{F \cdot \left(\frac{1}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)} \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{{B\_m}^{2} \cdot -0.5}{C}\right)\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \left(-\frac{F}{B\_m}\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= t_2 -5e-104)
     (*
      (sqrt
       (*
        F
        (*
         (/ 1.0 (fma -4.0 (* A C) (pow B_m 2.0)))
         (+ A (- C (hypot B_m (- A C)))))))
      (- (sqrt 2.0)))
     (if (<= t_2 INFINITY)
       (/
        (sqrt (* (* F t_0) (* 2.0 (+ A (+ A (/ (* (pow B_m 2.0) -0.5) C))))))
        (- t_0))
       (- (sqrt (* 2.0 (- (/ F B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -5e-104) {
		tmp = sqrt((F * ((1.0 / fma(-4.0, (A * C), pow(B_m, 2.0))) * (A + (C - hypot(B_m, (A - C))))))) * -sqrt(2.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + (A + ((pow(B_m, 2.0) * -0.5) / C)))))) / -t_0;
	} else {
		tmp = -sqrt((2.0 * -(F / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -5e-104)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(1.0 / fma(-4.0, Float64(A * C), (B_m ^ 2.0))) * Float64(A + Float64(C - hypot(B_m, Float64(A - C))))))) * Float64(-sqrt(2.0)));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(A + Float64(Float64((B_m ^ 2.0) * -0.5) / C)))))) / Float64(-t_0));
	else
		tmp = Float64(-sqrt(Float64(2.0 * Float64(-Float64(F / B_m)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-104], N[(N[Sqrt[N[(F * N[(N[(1.0 / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], (-N[Sqrt[N[(2.0 * (-N[(F / B$95$m), $MachinePrecision])), $MachinePrecision]], $MachinePrecision])]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.99999999999999979e-104

   1. Initial program 38.6%

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

   3. Taylor expanded in F around 0 44.2%

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

   5. Simplified 65.9%

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

      1. div-inv 65.9%

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

      2. associate--r- 66.6%

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

   7. Applied egg-rr 66.6%

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

   if -4.99999999999999979e-104 < (/.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 24.1%

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

   3. Simplified 33.2%

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

   5. Taylor expanded in C around inf 24.6%

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

      1. associate-*r/ 24.6%

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

      2. mul-1-neg 24.6%

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

   7. Simplified 24.6%

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

   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 F around 0 0.0%

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

   5. Simplified 4.0%

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

   6. Taylor expanded in B around inf 20.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-1}{B}}} \]
   7. Step-by-step derivation

      1. pow1 20.2%

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

      2. sqrt-unprod 20.2%

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

   8. Applied egg-rr 20.2%

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

      1. unpow1 20.2%

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

      2. associate-*r/ 20.3%

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

      3. *-commutative 20.3%

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

      4. mul-1-neg 20.3%

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

   10. Simplified 20.3%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{F \cdot \left(\frac{1}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)} \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{{B}^{2} \cdot -0.5}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \left(-\frac{F}{B}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := -\sqrt{2}\\ t_2 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_3 := -\sqrt{\left(2 \cdot F\right) \cdot \frac{\left(A + C\right) - t\_2}{\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}}\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ t_5 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_6 := \sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_1\\ \mathbf{if}\;{B\_m}^{2} \leq 1.5 \cdot 10^{-182}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_5\right) \cdot \left(2 \cdot \left(A + \left(C - t\_2\right)\right)\right)}}{-t\_5}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+33}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+74}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+215}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+249}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- (sqrt 2.0)))
        (t_2 (hypot B_m (- A C)))
        (t_3
         (-
          (sqrt
           (*
            (* 2.0 F)
            (/ (- (+ A C) t_2) (fma C (* A -4.0) (pow B_m 2.0)))))))
        (t_4
         (/
          (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
          (- t_0 (pow B_m 2.0))))
        (t_5 (fma B_m B_m (* A (* C -4.0))))
        (t_6 (* (sqrt (* F (/ -0.5 C))) t_1)))
   (if (<= (pow B_m 2.0) 1.5e-182)
     t_4
     (if (<= (pow B_m 2.0) 1e-23)
       (/ (sqrt (* (* F t_5) (* 2.0 (+ A (- C t_2))))) (- t_5))
       (if (<= (pow B_m 2.0) 1e+33)
         t_4
         (if (<= (pow B_m 2.0) 1e+74)
           t_6
           (if (<= (pow B_m 2.0) 1e+215)
             t_3
             (if (<= (pow B_m 2.0) 2e+249)
               t_6
               (if (<= (pow B_m 2.0) 4e+291)
                 t_3
                 (*
                  (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m)))
                  t_1))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = -sqrt(2.0);
	double t_2 = hypot(B_m, (A - C));
	double t_3 = -sqrt(((2.0 * F) * (((A + C) - t_2) / fma(C, (A * -4.0), pow(B_m, 2.0)))));
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	double t_5 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_6 = sqrt((F * (-0.5 / C))) * t_1;
	double tmp;
	if (pow(B_m, 2.0) <= 1.5e-182) {
		tmp = t_4;
	} else if (pow(B_m, 2.0) <= 1e-23) {
		tmp = sqrt(((F * t_5) * (2.0 * (A + (C - t_2))))) / -t_5;
	} else if (pow(B_m, 2.0) <= 1e+33) {
		tmp = t_4;
	} else if (pow(B_m, 2.0) <= 1e+74) {
		tmp = t_6;
	} else if (pow(B_m, 2.0) <= 1e+215) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 2e+249) {
		tmp = t_6;
	} else if (pow(B_m, 2.0) <= 4e+291) {
		tmp = t_3;
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_1;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(-sqrt(2.0))
	t_2 = hypot(B_m, Float64(A - C))
	t_3 = Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(Float64(Float64(A + C) - t_2) / fma(C, Float64(A * -4.0), (B_m ^ 2.0))))))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)))
	t_5 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_6 = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_1)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1.5e-182)
		tmp = t_4;
	elseif ((B_m ^ 2.0) <= 1e-23)
		tmp = Float64(sqrt(Float64(Float64(F * t_5) * Float64(2.0 * Float64(A + Float64(C - t_2))))) / Float64(-t_5));
	elseif ((B_m ^ 2.0) <= 1e+33)
		tmp = t_4;
	elseif ((B_m ^ 2.0) <= 1e+74)
		tmp = t_6;
	elseif ((B_m ^ 2.0) <= 1e+215)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 2e+249)
		tmp = t_6;
	elseif ((B_m ^ 2.0) <= 4e+291)
		tmp = t_3;
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * t_1);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$2 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(N[(A + C), $MachinePrecision] - t$95$2), $MachinePrecision] / N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1.5e-182], t$95$4, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-23], N[(N[Sqrt[N[(N[(F * t$95$5), $MachinePrecision] * N[(2.0 * N[(A + N[(C - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$5)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+33], t$95$4, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+74], t$95$6, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+215], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+249], t$95$6, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+291], t$95$3, N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.5000000000000001e-182 or 9.9999999999999996e-24 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999995e32

   1. Initial program 22.8%

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

   3. Taylor expanded in A around -inf 25.8%

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

   if 1.5000000000000001e-182 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999996e-24

   1. Initial program 41.3%

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

   3. Simplified 50.5%

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

   if 9.9999999999999995e32 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999952e73 or 9.99999999999999907e214 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999998e249

   1. Initial program 22.8%

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

   3. Taylor expanded in F around 0 16.1%

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

   5. Simplified 19.4%

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

   6. Taylor expanded in A around -inf 16.7%

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

   if 9.99999999999999952e73 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999907e214 or 1.9999999999999998e249 < (pow.f64 B #s(literal 2 binary64)) < 3.9999999999999998e291

   1. Initial program 22.9%

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

   3. Taylor expanded in F around 0 29.7%

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

   5. Simplified 59.4%

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

      1. add-sqr-sqrt 59.4%

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

      2. pow2 59.4%

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

      3. *-commutative 59.4%

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

      4. associate--r- 59.4%

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

   7. Applied egg-rr 59.4%

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

      1. pow1 59.4%

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

   9. Applied egg-rr 59.7%

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

       1. unpow1 59.7%

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

       2. unpow1/2 59.7%

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

       3. associate-*r* 59.7%

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

       4. fma-define 59.7%

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

       5. *-commutative 59.7%

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

       6. *-commutative 59.7%

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

       7. associate-*r* 59.7%

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

       8. fma-define 59.7%

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

       9. associate-+r- 59.5%

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

   11. Simplified 59.5%

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

   if 3.9999999999999998e291 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.1%

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

   3. Taylor expanded in F around 0 1.6%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in B around inf 33.0%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{\left(\frac{A}{B} + \frac{C}{B}\right) - 1}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 1.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+33}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+74}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{+215}:\\ \;\;\;\;-\sqrt{\left(2 \cdot F\right) \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+249}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;-\sqrt{\left(2 \cdot F\right) \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_2 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := -t\_2\\ t_4 := F \cdot t\_2\\ t_5 := -\sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 1.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-23}:\\ \;\;\;\;\frac{\sqrt{t\_4 \cdot \left(2 \cdot \left(A + \left(C - t\_1\right)\right)\right)}}{t\_3}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;-2 \cdot \sqrt{A \cdot \frac{F}{{B\_m}^{2} + -4 \cdot \left(A \cdot C\right)}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{F \cdot \frac{C + \left(A - t\_1\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot t\_5\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{t\_4 \cdot \left(2 \cdot \left(A + \left(A + \frac{{B\_m}^{2} \cdot -0.5}{C}\right)\right)\right)}}{t\_3}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+249}:\\ \;\;\;\;-\sqrt{\left(2 \cdot F\right) \cdot \frac{\left(A + C\right) - t\_1}{\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot t\_5\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (hypot B_m (- A C)))
        (t_2 (fma B_m B_m (* A (* C -4.0))))
        (t_3 (- t_2))
        (t_4 (* F t_2))
        (t_5 (- (sqrt 2.0))))
   (if (<= (pow B_m 2.0) 1.5e-182)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 1e-23)
       (/ (sqrt (* t_4 (* 2.0 (+ A (- C t_1))))) t_3)
       (if (<= (pow B_m 2.0) 5e-7)
         (* -2.0 (sqrt (* A (/ F (+ (pow B_m 2.0) (* -4.0 (* A C)))))))
         (if (<= (pow B_m 2.0) 2e+124)
           (*
            (sqrt (* F (/ (+ C (- A t_1)) (fma -4.0 (* A C) (pow B_m 2.0)))))
            t_5)
           (if (<= (pow B_m 2.0) 5e+136)
             (/
              (sqrt (* t_4 (* 2.0 (+ A (+ A (/ (* (pow B_m 2.0) -0.5) C))))))
              t_3)
             (if (<= (pow B_m 2.0) 2e+249)
               (-
                (sqrt
                 (*
                  (* 2.0 F)
                  (/ (- (+ A C) t_1) (fma C (* A -4.0) (pow B_m 2.0))))))
               (*
                (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m)))
                t_5)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = hypot(B_m, (A - C));
	double t_2 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_3 = -t_2;
	double t_4 = F * t_2;
	double t_5 = -sqrt(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 1.5e-182) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 1e-23) {
		tmp = sqrt((t_4 * (2.0 * (A + (C - t_1))))) / t_3;
	} else if (pow(B_m, 2.0) <= 5e-7) {
		tmp = -2.0 * sqrt((A * (F / (pow(B_m, 2.0) + (-4.0 * (A * C))))));
	} else if (pow(B_m, 2.0) <= 2e+124) {
		tmp = sqrt((F * ((C + (A - t_1)) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * t_5;
	} else if (pow(B_m, 2.0) <= 5e+136) {
		tmp = sqrt((t_4 * (2.0 * (A + (A + ((pow(B_m, 2.0) * -0.5) / C)))))) / t_3;
	} else if (pow(B_m, 2.0) <= 2e+249) {
		tmp = -sqrt(((2.0 * F) * (((A + C) - t_1) / fma(C, (A * -4.0), pow(B_m, 2.0)))));
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_5;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = hypot(B_m, Float64(A - C))
	t_2 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_3 = Float64(-t_2)
	t_4 = Float64(F * t_2)
	t_5 = Float64(-sqrt(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1.5e-182)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 1e-23)
		tmp = Float64(sqrt(Float64(t_4 * Float64(2.0 * Float64(A + Float64(C - t_1))))) / t_3);
	elseif ((B_m ^ 2.0) <= 5e-7)
		tmp = Float64(-2.0 * sqrt(Float64(A * Float64(F / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))))));
	elseif ((B_m ^ 2.0) <= 2e+124)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(C + Float64(A - t_1)) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * t_5);
	elseif ((B_m ^ 2.0) <= 5e+136)
		tmp = Float64(sqrt(Float64(t_4 * Float64(2.0 * Float64(A + Float64(A + Float64(Float64((B_m ^ 2.0) * -0.5) / C)))))) / t_3);
	elseif ((B_m ^ 2.0) <= 2e+249)
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(Float64(Float64(A + C) - t_1) / fma(C, Float64(A * -4.0), (B_m ^ 2.0))))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * t_5);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-t$95$2)}, Block[{t$95$4 = N[(F * t$95$2), $MachinePrecision]}, Block[{t$95$5 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1.5e-182], 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 * A), $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], 1e-23], N[(N[Sqrt[N[(t$95$4 * N[(2.0 * N[(A + N[(C - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-7], N[(-2.0 * N[Sqrt[N[(A * N[(F / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+124], N[(N[Sqrt[N[(F * N[(N[(C + N[(A - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+136], N[(N[Sqrt[N[(t$95$4 * N[(2.0 * N[(A + N[(A + N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+249], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(N[(A + C), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$5), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.5000000000000001e-182

   1. Initial program 21.1%

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

   3. Taylor expanded in A around -inf 26.3%

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

   if 1.5000000000000001e-182 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999996e-24

   1. Initial program 41.3%

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

   3. Simplified 50.5%

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

   if 9.9999999999999996e-24 < (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999977e-7

   1. Initial program 35.1%

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

   3. Taylor expanded in A around -inf 44.1%

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

   4. Taylor expanded in F around 0 44.7%

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

      1. associate-/l* 44.7%

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

      2. cancel-sign-sub-inv 44.7%

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

      3. metadata-eval 44.7%

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

   6. Simplified 44.7%

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

   if 4.99999999999999977e-7 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e124

   1. Initial program 30.0%

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

   3. Taylor expanded in F around 0 29.9%

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

   5. Simplified 45.4%

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

   if 1.9999999999999999e124 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000002e136

   1. Initial program 2.6%

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

   3. Simplified 3.7%

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

   5. Taylor expanded in C around inf 34.2%

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

      1. associate-*r/ 34.2%

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

      2. mul-1-neg 34.2%

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

   7. Simplified 34.2%

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

   if 5.0000000000000002e136 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999998e249

   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 23.1%

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

   5. Simplified 49.4%

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

      1. add-sqr-sqrt 49.3%

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

      2. pow2 49.3%

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

      3. *-commutative 49.3%

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

      4. associate--r- 50.2%

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

   7. Applied egg-rr 50.2%

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

      1. pow1 50.2%

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

   9. Applied egg-rr 50.4%

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

       1. unpow1 50.4%

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

       2. unpow1/2 50.4%

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

       3. associate-*r* 50.4%

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

       4. fma-define 50.4%

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

       5. *-commutative 50.4%

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

       6. *-commutative 50.4%

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

       7. associate-*r* 50.4%

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

       8. fma-define 50.4%

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

       9. associate-+r- 49.5%

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

   11. Simplified 49.5%

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

   if 1.9999999999999998e249 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 3.0%

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

   3. Taylor expanded in F around 0 7.9%

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

   5. Simplified 17.6%

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

   6. Taylor expanded in B around inf 32.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{\left(\frac{A}{B} + \frac{C}{B}\right) - 1}{B}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 1.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;-2 \cdot \sqrt{A \cdot \frac{F}{{B}^{2} + -4 \cdot \left(A \cdot C\right)}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{F \cdot \frac{C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{{B}^{2} \cdot -0.5}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+249}:\\ \;\;\;\;-\sqrt{\left(2 \cdot F\right) \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_2 := -\sqrt{\left(2 \cdot F\right) \cdot \frac{\left(A + C\right) - t\_1}{\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}}\\ t_3 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_4 := -t\_3\\ t_5 := F \cdot t\_3\\ \mathbf{if}\;{B\_m}^{2} \leq 1.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-23}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot \left(A + \left(C - t\_1\right)\right)\right)}}{t\_4}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;-2 \cdot \sqrt{A \cdot \frac{F}{{B\_m}^{2} + -4 \cdot \left(A \cdot C\right)}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+124}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot \left(A + \left(A + \frac{{B\_m}^{2} \cdot -0.5}{C}\right)\right)\right)}}{t\_4}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+249}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (hypot B_m (- A C)))
        (t_2
         (-
          (sqrt
           (*
            (* 2.0 F)
            (/ (- (+ A C) t_1) (fma C (* A -4.0) (pow B_m 2.0)))))))
        (t_3 (fma B_m B_m (* A (* C -4.0))))
        (t_4 (- t_3))
        (t_5 (* F t_3)))
   (if (<= (pow B_m 2.0) 1.5e-182)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 1e-23)
       (/ (sqrt (* t_5 (* 2.0 (+ A (- C t_1))))) t_4)
       (if (<= (pow B_m 2.0) 5e-7)
         (* -2.0 (sqrt (* A (/ F (+ (pow B_m 2.0) (* -4.0 (* A C)))))))
         (if (<= (pow B_m 2.0) 2e+124)
           t_2
           (if (<= (pow B_m 2.0) 5e+136)
             (/
              (sqrt (* t_5 (* 2.0 (+ A (+ A (/ (* (pow B_m 2.0) -0.5) C))))))
              t_4)
             (if (<= (pow B_m 2.0) 2e+249)
               t_2
               (*
                (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m)))
                (- (sqrt 2.0)))))))))))

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 = hypot(B_m, (A - C));
	double t_2 = -sqrt(((2.0 * F) * (((A + C) - t_1) / fma(C, (A * -4.0), pow(B_m, 2.0)))));
	double t_3 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_4 = -t_3;
	double t_5 = F * t_3;
	double tmp;
	if (pow(B_m, 2.0) <= 1.5e-182) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 1e-23) {
		tmp = sqrt((t_5 * (2.0 * (A + (C - t_1))))) / t_4;
	} else if (pow(B_m, 2.0) <= 5e-7) {
		tmp = -2.0 * sqrt((A * (F / (pow(B_m, 2.0) + (-4.0 * (A * C))))));
	} else if (pow(B_m, 2.0) <= 2e+124) {
		tmp = t_2;
	} else if (pow(B_m, 2.0) <= 5e+136) {
		tmp = sqrt((t_5 * (2.0 * (A + (A + ((pow(B_m, 2.0) * -0.5) / C)))))) / t_4;
	} else if (pow(B_m, 2.0) <= 2e+249) {
		tmp = t_2;
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * -sqrt(2.0);
	}
	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 = hypot(B_m, Float64(A - C))
	t_2 = Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(Float64(Float64(A + C) - t_1) / fma(C, Float64(A * -4.0), (B_m ^ 2.0))))))
	t_3 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_4 = Float64(-t_3)
	t_5 = Float64(F * t_3)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1.5e-182)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 1e-23)
		tmp = Float64(sqrt(Float64(t_5 * Float64(2.0 * Float64(A + Float64(C - t_1))))) / t_4);
	elseif ((B_m ^ 2.0) <= 5e-7)
		tmp = Float64(-2.0 * sqrt(Float64(A * Float64(F / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))))));
	elseif ((B_m ^ 2.0) <= 2e+124)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 5e+136)
		tmp = Float64(sqrt(Float64(t_5 * Float64(2.0 * Float64(A + Float64(A + Float64(Float64((B_m ^ 2.0) * -0.5) / C)))))) / t_4);
	elseif ((B_m ^ 2.0) <= 2e+249)
		tmp = t_2;
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * Float64(-sqrt(2.0)));
	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[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(N[(A + C), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$3 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-t$95$3)}, Block[{t$95$5 = N[(F * t$95$3), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1.5e-182], 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 * A), $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], 1e-23], N[(N[Sqrt[N[(t$95$5 * N[(2.0 * N[(A + N[(C - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-7], N[(-2.0 * N[Sqrt[N[(A * N[(F / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+124], t$95$2, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+136], N[(N[Sqrt[N[(t$95$5 * N[(2.0 * N[(A + N[(A + N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+249], t$95$2, N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.5000000000000001e-182

   1. Initial program 21.1%

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

   3. Taylor expanded in A around -inf 26.3%

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

   if 1.5000000000000001e-182 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999996e-24

   1. Initial program 41.3%

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

   3. Simplified 50.5%

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

   if 9.9999999999999996e-24 < (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999977e-7

   1. Initial program 35.1%

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

   3. Taylor expanded in A around -inf 44.1%

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

   4. Taylor expanded in F around 0 44.7%

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

      1. associate-/l* 44.7%

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

      2. cancel-sign-sub-inv 44.7%

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

      3. metadata-eval 44.7%

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

   6. Simplified 44.7%

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

   if 4.99999999999999977e-7 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e124 or 5.0000000000000002e136 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999998e249

   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 27.2%

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

   5. Simplified 47.0%

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

      1. add-sqr-sqrt 46.9%

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

      2. pow2 46.9%

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

      3. *-commutative 46.9%

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

      4. associate--r- 47.1%

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

   7. Applied egg-rr 47.1%

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

      1. pow1 47.1%

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

   9. Applied egg-rr 47.2%

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

       1. unpow1 47.2%

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

       2. unpow1/2 47.2%

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

       3. associate-*r* 47.2%

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

       4. fma-define 47.2%

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

       5. *-commutative 47.2%

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

       6. *-commutative 47.2%

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

       7. associate-*r* 47.2%

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

       8. fma-define 47.2%

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

       9. associate-+r- 46.8%

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

   11. Simplified 46.8%

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

   if 1.9999999999999999e124 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000002e136

   1. Initial program 2.6%

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

   3. Simplified 3.7%

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

   5. Taylor expanded in C around inf 34.2%

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

      1. associate-*r/ 34.2%

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

      2. mul-1-neg 34.2%

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

   7. Simplified 34.2%

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

   if 1.9999999999999998e249 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 3.0%

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

   3. Taylor expanded in F around 0 7.9%

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

   5. Simplified 17.6%

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

   6. Taylor expanded in B around inf 32.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{\left(\frac{A}{B} + \frac{C}{B}\right) - 1}{B}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 1.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;-2 \cdot \sqrt{A \cdot \frac{F}{{B}^{2} + -4 \cdot \left(A \cdot C\right)}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+124}:\\ \;\;\;\;-\sqrt{\left(2 \cdot F\right) \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{{B}^{2} \cdot -0.5}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+249}:\\ \;\;\;\;-\sqrt{\left(2 \cdot F\right) \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - {B\_m}^{2}\\ t_1 := -\sqrt{2}\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ t_3 := \sqrt{-8 \cdot \left(\left(A + A\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}\\ t_4 := B\_m \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-59}:\\ \;\;\;\;\frac{t\_3}{-t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{t\_4}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;t\_3 \cdot \frac{-1}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{t\_4}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* (* 4.0 A) C) (pow B_m 2.0)))
        (t_1 (- (sqrt 2.0)))
        (t_2 (fma C (* A -4.0) (pow B_m 2.0)))
        (t_3 (sqrt (* -8.0 (* (+ A A) (* F (* A C))))))
        (t_4 (* B_m (sqrt (* 2.0 (* F (- A (hypot B_m A))))))))
   (if (<= (pow B_m 2.0) 2e-59)
     (/ t_3 (- t_2))
     (if (<= (pow B_m 2.0) 5e-20)
       (/ t_4 t_0)
       (if (<= (pow B_m 2.0) 2e+26)
         (* t_3 (/ -1.0 t_2))
         (if (<= (pow B_m 2.0) 2e+94)
           (* (sqrt (* F (/ -0.5 C))) t_1)
           (if (<= (pow B_m 2.0) 4e+291)
             (/ 1.0 (/ t_0 t_4))
             (*
              (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m)))
              t_1))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - pow(B_m, 2.0);
	double t_1 = -sqrt(2.0);
	double t_2 = fma(C, (A * -4.0), pow(B_m, 2.0));
	double t_3 = sqrt((-8.0 * ((A + A) * (F * (A * C)))));
	double t_4 = B_m * sqrt((2.0 * (F * (A - hypot(B_m, A)))));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-59) {
		tmp = t_3 / -t_2;
	} else if (pow(B_m, 2.0) <= 5e-20) {
		tmp = t_4 / t_0;
	} else if (pow(B_m, 2.0) <= 2e+26) {
		tmp = t_3 * (-1.0 / t_2);
	} else if (pow(B_m, 2.0) <= 2e+94) {
		tmp = sqrt((F * (-0.5 / C))) * t_1;
	} else if (pow(B_m, 2.0) <= 4e+291) {
		tmp = 1.0 / (t_0 / t_4);
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_1;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0))
	t_1 = Float64(-sqrt(2.0))
	t_2 = fma(C, Float64(A * -4.0), (B_m ^ 2.0))
	t_3 = sqrt(Float64(-8.0 * Float64(Float64(A + A) * Float64(F * Float64(A * C)))))
	t_4 = Float64(B_m * sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-59)
		tmp = Float64(t_3 / Float64(-t_2));
	elseif ((B_m ^ 2.0) <= 5e-20)
		tmp = Float64(t_4 / t_0);
	elseif ((B_m ^ 2.0) <= 2e+26)
		tmp = Float64(t_3 * Float64(-1.0 / t_2));
	elseif ((B_m ^ 2.0) <= 2e+94)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_1);
	elseif ((B_m ^ 2.0) <= 4e+291)
		tmp = Float64(1.0 / Float64(t_0 / t_4));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * t_1);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$2 = N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(-8.0 * N[(N[(A + A), $MachinePrecision] * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(B$95$m * N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-59], N[(t$95$3 / (-t$95$2)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-20], N[(t$95$4 / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+26], N[(t$95$3 * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+94], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+291], N[(1.0 / N[(t$95$0 / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-59

   1. Initial program 24.8%

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

   3. Simplified 29.0%

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

   5. Taylor expanded in C around inf 19.2%

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

      1. associate-*r* 20.0%

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

      2. *-commutative 20.0%

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

      3. mul-1-neg 20.0%

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

   7. Simplified 20.0%

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

      1. *-un-lft-identity 20.0%

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

      2. associate-*r* 21.4%

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

   9. Applied egg-rr 21.4%

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

       1. *-lft-identity 21.4%

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

   11. Simplified 21.4%

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

   if 2.0000000000000001e-59 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999999e-20

   1. Initial program 71.9%

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

   3. Taylor expanded in C around 0 2.9%

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

      1. associate-*l* 2.9%

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

      2. +-commutative 2.9%

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

      3. unpow2 2.9%

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

      4. unpow2 2.9%

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

      5. hypot-define 2.9%

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

   5. Simplified 2.9%

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

      1. *-un-lft-identity 2.9%

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

      2. distribute-rgt-neg-in 2.9%

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

      3. sqrt-unprod 2.9%

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

      4. *-commutative 2.9%

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

      5. *-commutative 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. *-lft-identity 2.9%

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

   9. Simplified 2.9%

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

   if 4.9999999999999999e-20 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e26

   1. Initial program 30.1%

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

   3. Simplified 33.1%

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

   5. Taylor expanded in C around inf 16.1%

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

      1. associate-*r* 16.1%

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

      2. *-commutative 16.1%

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

      3. mul-1-neg 16.1%

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

   7. Simplified 16.1%

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

      1. div-inv 16.3%

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

      2. associate-*r* 16.6%

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

   9. Applied egg-rr 16.6%

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

   if 2.0000000000000001e26 < (pow.f64 B #s(literal 2 binary64)) < 2e94

   1. Initial program 23.2%

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

   3. Taylor expanded in F around 0 23.1%

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

   5. Simplified 32.5%

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

   6. Taylor expanded in A around -inf 23.9%

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

   if 2e94 < (pow.f64 B #s(literal 2 binary64)) < 3.9999999999999998e291

   1. Initial program 23.6%

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

   3. Taylor expanded in C around 0 19.5%

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

      1. associate-*l* 19.5%

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

      2. +-commutative 19.5%

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

      3. unpow2 19.5%

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

      4. unpow2 19.5%

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

      5. hypot-define 26.6%

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

   5. Simplified 26.6%

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

      1. clear-num 26.6%

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

      2. inv-pow 26.6%

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

      3. *-commutative 26.6%

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

      4. *-commutative 26.6%

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

      5. distribute-rgt-neg-in 26.6%

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

      6. sqrt-unprod 26.7%

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

   7. Applied egg-rr 26.7%

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

      1. unpow-1 26.7%

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

   9. Simplified 26.7%

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

   if 3.9999999999999998e291 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.1%

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

   3. Taylor expanded in F around 0 1.6%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in B around inf 33.0%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{\left(\frac{A}{B} + \frac{C}{B}\right) - 1}{B}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A + A\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{B \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{-8 \cdot \left(\left(A + A\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\frac{1}{\frac{\left(4 \cdot A\right) \cdot C - {B}^{2}}{B \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := -\sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+33}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{hypot}\left(B\_m, C\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+212}:\\ \;\;\;\;-\sqrt{\left(2 \cdot F\right) \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)) (t_1 (- (sqrt 2.0))))
   (if (<= (pow B_m 2.0) 1e+33)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 5e+139)
       (* (sqrt (* F (/ -0.5 C))) t_1)
       (if (<= (pow B_m 2.0) 5e+177)
         (* (sqrt (* F (- C (hypot B_m C)))) (/ (sqrt 2.0) (- B_m)))
         (if (<= (pow B_m 2.0) 5e+212)
           (-
            (sqrt
             (*
              (* 2.0 F)
              (/
               (- (+ A C) (hypot B_m (- A C)))
               (fma C (* A -4.0) (pow B_m 2.0))))))
           (* (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m))) t_1)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = -sqrt(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 1e+33) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 5e+139) {
		tmp = sqrt((F * (-0.5 / C))) * t_1;
	} else if (pow(B_m, 2.0) <= 5e+177) {
		tmp = sqrt((F * (C - hypot(B_m, C)))) * (sqrt(2.0) / -B_m);
	} else if (pow(B_m, 2.0) <= 5e+212) {
		tmp = -sqrt(((2.0 * F) * (((A + C) - hypot(B_m, (A - C))) / fma(C, (A * -4.0), pow(B_m, 2.0)))));
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_1;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(-sqrt(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+33)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 5e+139)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_1);
	elseif ((B_m ^ 2.0) <= 5e+177)
		tmp = Float64(sqrt(Float64(F * Float64(C - hypot(B_m, C)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	elseif ((B_m ^ 2.0) <= 5e+212)
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(Float64(Float64(A + C) - hypot(B_m, Float64(A - C))) / fma(C, Float64(A * -4.0), (B_m ^ 2.0))))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * t_1);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+33], 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 * A), $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+139], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+177], 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], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+212], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999995e32

   1. Initial program 27.8%

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

   3. Taylor expanded in A around -inf 21.5%

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

   if 9.9999999999999995e32 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e139

   1. Initial program 20.9%

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

   3. Taylor expanded in F around 0 21.0%

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

   5. Simplified 34.2%

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

   6. Taylor expanded in A around -inf 23.6%

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

   if 5.0000000000000003e139 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e177

   1. Initial program 60.6%

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

   3. Taylor expanded in A around 0 40.5%

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

      1. mul-1-neg 40.5%

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

      2. unpow2 40.5%

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

      3. unpow2 40.5%

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

      4. hypot-define 41.5%

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

   5. Simplified 41.5%

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

   if 5.0000000000000003e177 < (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999992e212

   1. Initial program 3.0%

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

   3. Taylor expanded in F around 0 18.9%

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

   5. Simplified 83.1%

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

      1. add-sqr-sqrt 82.8%

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

      2. pow2 82.8%

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

      3. *-commutative 82.8%

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

      4. associate--r- 82.8%

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

   7. Applied egg-rr 82.8%

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

      1. pow1 82.8%

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

   9. Applied egg-rr 83.1%

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

       1. unpow1 83.1%

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

       2. unpow1/2 83.1%

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

       3. associate-*r* 83.1%

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

       4. fma-define 83.1%

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

       5. *-commutative 83.1%

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

       6. *-commutative 83.1%

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

       7. associate-*r* 83.1%

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

       8. fma-define 83.1%

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

       9. associate-+r- 83.1%

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

   11. Simplified 83.1%

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

   if 4.99999999999999992e212 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 5.2%

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

   3. Taylor expanded in F around 0 8.6%

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

   5. Simplified 18.8%

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

   6. Taylor expanded in B around inf 30.0%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{\left(\frac{A}{B} + \frac{C}{B}\right) - 1}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+33}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+212}:\\ \;\;\;\;-\sqrt{\left(2 \cdot F\right) \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\\ t_1 := -\sqrt{2}\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ t_3 := \sqrt{-8 \cdot \left(\left(A + A\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-59}:\\ \;\;\;\;\frac{t\_3}{-t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{B\_m \cdot \sqrt{2 \cdot t\_0}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;t\_3 \cdot \frac{-1}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* F (- A (hypot B_m A))))
        (t_1 (- (sqrt 2.0)))
        (t_2 (fma C (* A -4.0) (pow B_m 2.0)))
        (t_3 (sqrt (* -8.0 (* (+ A A) (* F (* A C)))))))
   (if (<= (pow B_m 2.0) 2e-59)
     (/ t_3 (- t_2))
     (if (<= (pow B_m 2.0) 5e-20)
       (/ (* B_m (sqrt (* 2.0 t_0))) (- (* (* 4.0 A) C) (pow B_m 2.0)))
       (if (<= (pow B_m 2.0) 2e+26)
         (* t_3 (/ -1.0 t_2))
         (if (<= (pow B_m 2.0) 2e+94)
           (* (sqrt (* F (/ -0.5 C))) t_1)
           (if (<= (pow B_m 2.0) 4e+291)
             (* (/ (sqrt 2.0) B_m) (- (sqrt t_0)))
             (*
              (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m)))
              t_1))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = F * (A - hypot(B_m, A));
	double t_1 = -sqrt(2.0);
	double t_2 = fma(C, (A * -4.0), pow(B_m, 2.0));
	double t_3 = sqrt((-8.0 * ((A + A) * (F * (A * C)))));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-59) {
		tmp = t_3 / -t_2;
	} else if (pow(B_m, 2.0) <= 5e-20) {
		tmp = (B_m * sqrt((2.0 * t_0))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+26) {
		tmp = t_3 * (-1.0 / t_2);
	} else if (pow(B_m, 2.0) <= 2e+94) {
		tmp = sqrt((F * (-0.5 / C))) * t_1;
	} else if (pow(B_m, 2.0) <= 4e+291) {
		tmp = (sqrt(2.0) / B_m) * -sqrt(t_0);
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_1;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(F * Float64(A - hypot(B_m, A)))
	t_1 = Float64(-sqrt(2.0))
	t_2 = fma(C, Float64(A * -4.0), (B_m ^ 2.0))
	t_3 = sqrt(Float64(-8.0 * Float64(Float64(A + A) * Float64(F * Float64(A * C)))))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-59)
		tmp = Float64(t_3 / Float64(-t_2));
	elseif ((B_m ^ 2.0) <= 5e-20)
		tmp = Float64(Float64(B_m * sqrt(Float64(2.0 * t_0))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+26)
		tmp = Float64(t_3 * Float64(-1.0 / t_2));
	elseif ((B_m ^ 2.0) <= 2e+94)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_1);
	elseif ((B_m ^ 2.0) <= 4e+291)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(t_0)));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * t_1);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$2 = N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(-8.0 * N[(N[(A + A), $MachinePrecision] * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-59], N[(t$95$3 / (-t$95$2)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-20], N[(N[(B$95$m * N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+26], N[(t$95$3 * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+94], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+291], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[t$95$0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-59

   1. Initial program 24.8%

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

   3. Simplified 29.0%

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

   5. Taylor expanded in C around inf 19.2%

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

      1. associate-*r* 20.0%

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

      2. *-commutative 20.0%

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

      3. mul-1-neg 20.0%

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

   7. Simplified 20.0%

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

      1. *-un-lft-identity 20.0%

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

      2. associate-*r* 21.4%

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

   9. Applied egg-rr 21.4%

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

       1. *-lft-identity 21.4%

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

   11. Simplified 21.4%

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

   if 2.0000000000000001e-59 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999999e-20

   1. Initial program 71.9%

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

   3. Taylor expanded in C around 0 2.9%

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

      1. associate-*l* 2.9%

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

      2. +-commutative 2.9%

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

      3. unpow2 2.9%

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

      4. unpow2 2.9%

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

      5. hypot-define 2.9%

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

   5. Simplified 2.9%

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

      1. *-un-lft-identity 2.9%

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

      2. distribute-rgt-neg-in 2.9%

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

      3. sqrt-unprod 2.9%

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

      4. *-commutative 2.9%

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

      5. *-commutative 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. *-lft-identity 2.9%

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

   9. Simplified 2.9%

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

   if 4.9999999999999999e-20 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e26

   1. Initial program 30.1%

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

   3. Simplified 33.1%

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

   5. Taylor expanded in C around inf 16.1%

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

      1. associate-*r* 16.1%

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

      2. *-commutative 16.1%

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

      3. mul-1-neg 16.1%

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

   7. Simplified 16.1%

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

      1. div-inv 16.3%

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

      2. associate-*r* 16.6%

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

   9. Applied egg-rr 16.6%

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

   if 2.0000000000000001e26 < (pow.f64 B #s(literal 2 binary64)) < 2e94

   1. Initial program 23.2%

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

   3. Taylor expanded in F around 0 23.1%

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

   5. Simplified 32.5%

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

   6. Taylor expanded in A around -inf 23.9%

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

   if 2e94 < (pow.f64 B #s(literal 2 binary64)) < 3.9999999999999998e291

   1. Initial program 23.6%

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

   3. Taylor expanded in C around 0 19.6%

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

      1. mul-1-neg 19.6%

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

      2. +-commutative 19.6%

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

      3. unpow2 19.6%

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

      4. unpow2 19.6%

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

      5. hypot-define 26.6%

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

   5. Simplified 26.6%

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

   if 3.9999999999999998e291 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.1%

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

   3. Taylor expanded in F around 0 1.6%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in B around inf 33.0%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{\left(\frac{A}{B} + \frac{C}{B}\right) - 1}{B}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A + A\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{B \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{-8 \cdot \left(\left(A + A\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\\ t_1 := -\sqrt{2}\\ t_2 := \frac{\sqrt{-8 \cdot \left(\left(A + A\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-59}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{B\_m \cdot \sqrt{2 \cdot t\_0}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* F (- A (hypot B_m A))))
        (t_1 (- (sqrt 2.0)))
        (t_2
         (/
          (sqrt (* -8.0 (* (+ A A) (* F (* A C)))))
          (- (fma C (* A -4.0) (pow B_m 2.0))))))
   (if (<= (pow B_m 2.0) 2e-59)
     t_2
     (if (<= (pow B_m 2.0) 5e-20)
       (/ (* B_m (sqrt (* 2.0 t_0))) (- (* (* 4.0 A) C) (pow B_m 2.0)))
       (if (<= (pow B_m 2.0) 2e+26)
         t_2
         (if (<= (pow B_m 2.0) 2e+94)
           (* (sqrt (* F (/ -0.5 C))) t_1)
           (if (<= (pow B_m 2.0) 4e+291)
             (* (/ (sqrt 2.0) B_m) (- (sqrt t_0)))
             (*
              (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m)))
              t_1))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = F * (A - hypot(B_m, A));
	double t_1 = -sqrt(2.0);
	double t_2 = sqrt((-8.0 * ((A + A) * (F * (A * C))))) / -fma(C, (A * -4.0), pow(B_m, 2.0));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-59) {
		tmp = t_2;
	} else if (pow(B_m, 2.0) <= 5e-20) {
		tmp = (B_m * sqrt((2.0 * t_0))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+26) {
		tmp = t_2;
	} else if (pow(B_m, 2.0) <= 2e+94) {
		tmp = sqrt((F * (-0.5 / C))) * t_1;
	} else if (pow(B_m, 2.0) <= 4e+291) {
		tmp = (sqrt(2.0) / B_m) * -sqrt(t_0);
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_1;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(F * Float64(A - hypot(B_m, A)))
	t_1 = Float64(-sqrt(2.0))
	t_2 = Float64(sqrt(Float64(-8.0 * Float64(Float64(A + A) * Float64(F * Float64(A * C))))) / Float64(-fma(C, Float64(A * -4.0), (B_m ^ 2.0))))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-59)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 5e-20)
		tmp = Float64(Float64(B_m * sqrt(Float64(2.0 * t_0))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+26)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 2e+94)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_1);
	elseif ((B_m ^ 2.0) <= 4e+291)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(t_0)));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * t_1);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$2 = N[(N[Sqrt[N[(-8.0 * N[(N[(A + A), $MachinePrecision] * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-59], t$95$2, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-20], N[(N[(B$95$m * N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+26], t$95$2, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+94], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+291], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[t$95$0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-59 or 4.9999999999999999e-20 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e26

   1. Initial program 25.1%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in C around inf 19.1%

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

      1. associate-*r* 19.8%

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

      2. *-commutative 19.8%

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

      3. mul-1-neg 19.8%

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

   7. Simplified 19.8%

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

      1. *-un-lft-identity 19.8%

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

      2. associate-*r* 21.1%

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

   9. Applied egg-rr 21.1%

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

       1. *-lft-identity 21.1%

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

   11. Simplified 21.1%

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

   if 2.0000000000000001e-59 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999999e-20

   1. Initial program 71.9%

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

   3. Taylor expanded in C around 0 2.9%

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

      1. associate-*l* 2.9%

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

      2. +-commutative 2.9%

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

      3. unpow2 2.9%

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

      4. unpow2 2.9%

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

      5. hypot-define 2.9%

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

   5. Simplified 2.9%

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

      1. *-un-lft-identity 2.9%

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

      2. distribute-rgt-neg-in 2.9%

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

      3. sqrt-unprod 2.9%

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

      4. *-commutative 2.9%

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

      5. *-commutative 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. *-lft-identity 2.9%

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

   9. Simplified 2.9%

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

   if 2.0000000000000001e26 < (pow.f64 B #s(literal 2 binary64)) < 2e94

   1. Initial program 23.2%

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

   3. Taylor expanded in F around 0 23.1%

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

   5. Simplified 32.5%

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

   6. Taylor expanded in A around -inf 23.9%

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

   if 2e94 < (pow.f64 B #s(literal 2 binary64)) < 3.9999999999999998e291

   1. Initial program 23.6%

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

   3. Taylor expanded in C around 0 19.6%

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

      1. mul-1-neg 19.6%

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

      2. +-commutative 19.6%

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

      3. unpow2 19.6%

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

      4. unpow2 19.6%

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

      5. hypot-define 26.6%

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

   5. Simplified 26.6%

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

   if 3.9999999999999998e291 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.1%

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

   3. Taylor expanded in F around 0 1.6%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in B around inf 33.0%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{\left(\frac{A}{B} + \frac{C}{B}\right) - 1}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A + A\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{B \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A + A\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{-8 \cdot \left(\left(A + A\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}\\ t_1 := \frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ t_2 := -\sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_2\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot t\_2\\ \end{array} \end{array} \]

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 (* -8.0 (* (+ A A) (* F (* A C)))))
          (- (fma C (* A -4.0) (pow B_m 2.0)))))
        (t_1 (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (- A (hypot B_m A)))))))
        (t_2 (- (sqrt 2.0))))
   (if (<= (pow B_m 2.0) 2e-59)
     t_0
     (if (<= (pow B_m 2.0) 5e-20)
       t_1
       (if (<= (pow B_m 2.0) 2e+26)
         t_0
         (if (<= (pow B_m 2.0) 2e+94)
           (* (sqrt (* F (/ -0.5 C))) t_2)
           (if (<= (pow B_m 2.0) 4e+291)
             t_1
             (*
              (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m)))
              t_2))))))))

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((-8.0 * ((A + A) * (F * (A * C))))) / -fma(C, (A * -4.0), pow(B_m, 2.0));
	double t_1 = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	double t_2 = -sqrt(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-59) {
		tmp = t_0;
	} else if (pow(B_m, 2.0) <= 5e-20) {
		tmp = t_1;
	} else if (pow(B_m, 2.0) <= 2e+26) {
		tmp = t_0;
	} else if (pow(B_m, 2.0) <= 2e+94) {
		tmp = sqrt((F * (-0.5 / C))) * t_2;
	} else if (pow(B_m, 2.0) <= 4e+291) {
		tmp = t_1;
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_2;
	}
	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(-8.0 * Float64(Float64(A + A) * Float64(F * Float64(A * C))))) / Float64(-fma(C, Float64(A * -4.0), (B_m ^ 2.0))))
	t_1 = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))))
	t_2 = Float64(-sqrt(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-59)
		tmp = t_0;
	elseif ((B_m ^ 2.0) <= 5e-20)
		tmp = t_1;
	elseif ((B_m ^ 2.0) <= 2e+26)
		tmp = t_0;
	elseif ((B_m ^ 2.0) <= 2e+94)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_2);
	elseif ((B_m ^ 2.0) <= 4e+291)
		tmp = t_1;
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * t_2);
	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[Sqrt[N[(-8.0 * N[(N[(A + A), $MachinePrecision] * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-59], t$95$0, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-20], t$95$1, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+26], t$95$0, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+94], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+291], t$95$1, N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-59 or 4.9999999999999999e-20 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e26

   1. Initial program 25.1%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in C around inf 19.1%

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

      1. associate-*r* 19.8%

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

      2. *-commutative 19.8%

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

      3. mul-1-neg 19.8%

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

   7. Simplified 19.8%

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

      1. *-un-lft-identity 19.8%

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

      2. associate-*r* 21.1%

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

   9. Applied egg-rr 21.1%

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

       1. *-lft-identity 21.1%

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

   11. Simplified 21.1%

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

   if 2.0000000000000001e-59 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999999e-20 or 2e94 < (pow.f64 B #s(literal 2 binary64)) < 3.9999999999999998e291

   1. Initial program 30.0%

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

   3. Taylor expanded in C around 0 17.4%

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

      1. mul-1-neg 17.4%

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

      2. +-commutative 17.4%

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

      3. unpow2 17.4%

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

      4. unpow2 17.4%

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

      5. hypot-define 23.5%

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

   5. Simplified 23.5%

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

   if 2.0000000000000001e26 < (pow.f64 B #s(literal 2 binary64)) < 2e94

   1. Initial program 23.2%

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

   3. Taylor expanded in F around 0 23.1%

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

   5. Simplified 32.5%

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

   6. Taylor expanded in A around -inf 23.9%

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

   if 3.9999999999999998e291 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.1%

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

   3. Taylor expanded in F around 0 1.6%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in B around inf 33.0%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{\left(\frac{A}{B} + \frac{C}{B}\right) - 1}{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A + A\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A + A\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := -\sqrt{2}\\ t_2 := t\_0 - {B\_m}^{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-123}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+63}:\\ \;\;\;\;\sqrt{F \cdot \frac{C + \left(A - \mathsf{hypot}\left(B\_m, A - C\right)\right)}{{B\_m}^{2}}} \cdot t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.25 \cdot \frac{A + A}{A \cdot C}\right)} \cdot t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+195}:\\ \;\;\;\;\left(B\_m \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}\right) \cdot \frac{1}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)) (t_1 (- (sqrt 2.0))) (t_2 (- t_0 (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 1e-123)
     (/ (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A))) t_2)
     (if (<= (pow B_m 2.0) 1e+63)
       (* (sqrt (* F (/ (+ C (- A (hypot B_m (- A C)))) (pow B_m 2.0)))) t_1)
       (if (<= (pow B_m 2.0) 5e+139)
         (* (sqrt (* F (* -0.25 (/ (+ A A) (* A C))))) t_1)
         (if (<= (pow B_m 2.0) 1e+195)
           (* (* B_m (sqrt (* 2.0 (* F (- A (hypot B_m A)))))) (/ 1.0 t_2))
           (if (<= (pow B_m 2.0) 5e+199)
             (* (sqrt (* F (/ -0.5 C))) t_1)
             (*
              (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m)))
              t_1))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = -sqrt(2.0);
	double t_2 = t_0 - pow(B_m, 2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 1e-123) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_2;
	} else if (pow(B_m, 2.0) <= 1e+63) {
		tmp = sqrt((F * ((C + (A - hypot(B_m, (A - C)))) / pow(B_m, 2.0)))) * t_1;
	} else if (pow(B_m, 2.0) <= 5e+139) {
		tmp = sqrt((F * (-0.25 * ((A + A) / (A * C))))) * t_1;
	} else if (pow(B_m, 2.0) <= 1e+195) {
		tmp = (B_m * sqrt((2.0 * (F * (A - hypot(B_m, A)))))) * (1.0 / t_2);
	} else if (pow(B_m, 2.0) <= 5e+199) {
		tmp = sqrt((F * (-0.5 / C))) * t_1;
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_1;
	}
	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);
	double t_2 = t_0 - Math.pow(B_m, 2.0);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-123) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_2;
	} else if (Math.pow(B_m, 2.0) <= 1e+63) {
		tmp = Math.sqrt((F * ((C + (A - Math.hypot(B_m, (A - C)))) / Math.pow(B_m, 2.0)))) * t_1;
	} else if (Math.pow(B_m, 2.0) <= 5e+139) {
		tmp = Math.sqrt((F * (-0.25 * ((A + A) / (A * C))))) * t_1;
	} else if (Math.pow(B_m, 2.0) <= 1e+195) {
		tmp = (B_m * Math.sqrt((2.0 * (F * (A - Math.hypot(B_m, A)))))) * (1.0 / t_2);
	} else if (Math.pow(B_m, 2.0) <= 5e+199) {
		tmp = Math.sqrt((F * (-0.5 / C))) * t_1;
	} else {
		tmp = Math.sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_1;
	}
	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)
	t_2 = t_0 - math.pow(B_m, 2.0)
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-123:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_2
	elif math.pow(B_m, 2.0) <= 1e+63:
		tmp = math.sqrt((F * ((C + (A - math.hypot(B_m, (A - C)))) / math.pow(B_m, 2.0)))) * t_1
	elif math.pow(B_m, 2.0) <= 5e+139:
		tmp = math.sqrt((F * (-0.25 * ((A + A) / (A * C))))) * t_1
	elif math.pow(B_m, 2.0) <= 1e+195:
		tmp = (B_m * math.sqrt((2.0 * (F * (A - math.hypot(B_m, A)))))) * (1.0 / t_2)
	elif math.pow(B_m, 2.0) <= 5e+199:
		tmp = math.sqrt((F * (-0.5 / C))) * t_1
	else:
		tmp = math.sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_1
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(-sqrt(2.0))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-123)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / t_2);
	elseif ((B_m ^ 2.0) <= 1e+63)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(C + Float64(A - hypot(B_m, Float64(A - C)))) / (B_m ^ 2.0)))) * t_1);
	elseif ((B_m ^ 2.0) <= 5e+139)
		tmp = Float64(sqrt(Float64(F * Float64(-0.25 * Float64(Float64(A + A) / Float64(A * C))))) * t_1);
	elseif ((B_m ^ 2.0) <= 1e+195)
		tmp = Float64(Float64(B_m * sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A)))))) * Float64(1.0 / t_2));
	elseif ((B_m ^ 2.0) <= 5e+199)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_1);
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * t_1);
	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);
	t_2 = t_0 - (B_m ^ 2.0);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-123)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / t_2;
	elseif ((B_m ^ 2.0) <= 1e+63)
		tmp = sqrt((F * ((C + (A - hypot(B_m, (A - C)))) / (B_m ^ 2.0)))) * t_1;
	elseif ((B_m ^ 2.0) <= 5e+139)
		tmp = sqrt((F * (-0.25 * ((A + A) / (A * C))))) * t_1;
	elseif ((B_m ^ 2.0) <= 1e+195)
		tmp = (B_m * sqrt((2.0 * (F * (A - hypot(B_m, A)))))) * (1.0 / t_2);
	elseif ((B_m ^ 2.0) <= 5e+199)
		tmp = sqrt((F * (-0.5 / C))) * t_1;
	else
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_1;
	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[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-123], 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 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+63], N[(N[Sqrt[N[(F * N[(N[(C + N[(A - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+139], N[(N[Sqrt[N[(F * N[(-0.25 * N[(N[(A + A), $MachinePrecision] / N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+195], N[(N[(B$95$m * N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+199], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.0000000000000001e-123

   1. Initial program 25.4%

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

   3. Taylor expanded in A around -inf 23.4%

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

   if 1.0000000000000001e-123 < (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000006e63

   1. Initial program 35.4%

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

   3. Taylor expanded in F around 0 35.3%

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

   5. Simplified 45.4%

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

   6. Taylor expanded in C around 0 43.5%

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

   if 1.00000000000000006e63 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e139

   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 17.8%

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

   5. Simplified 35.3%

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

   6. Taylor expanded in C around inf 26.4%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\left(-0.25 \cdot \frac{A - -1 \cdot A}{A \cdot C}\right)}} \]
   7. Step-by-step derivation

      1. neg-mul-1 26.4%

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

   8. Simplified 26.4%

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

   if 5.0000000000000003e139 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999977e194

   1. Initial program 38.9%

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

   3. Taylor expanded in C around 0 38.9%

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

      1. associate-*l* 38.8%

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

      2. +-commutative 38.8%

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

      3. unpow2 38.8%

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

      4. unpow2 38.8%

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

      5. hypot-define 62.9%

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

   5. Simplified 62.9%

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

      1. div-inv 63.1%

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

      2. distribute-rgt-neg-in 63.1%

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

      3. sqrt-unprod 63.4%

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

      4. *-commutative 63.4%

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

      5. *-commutative 63.4%

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

   7. Applied egg-rr 63.4%

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

   if 9.99999999999999977e194 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999998e199

   1. Initial program 3.1%

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

   3. Taylor expanded in F around 0 3.1%

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

   5. Simplified 51.6%

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

   6. Taylor expanded in A around -inf 50.8%

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

   if 4.9999999999999998e199 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 5.2%

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

   3. Taylor expanded in F around 0 8.6%

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

   5. Simplified 19.7%

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

   6. Taylor expanded in B around inf 29.9%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{\left(\frac{A}{B} + \frac{C}{B}\right) - 1}{B}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-123}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+63}:\\ \;\;\;\;\sqrt{F \cdot \frac{C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)}{{B}^{2}}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.25 \cdot \frac{A + A}{A \cdot C}\right)} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{+195}:\\ \;\;\;\;\left(B \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}\right) \cdot \frac{1}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := -\sqrt{2}\\ t_3 := \sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_2\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+33}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+139}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+195}:\\ \;\;\;\;\left(B\_m \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}\right) \cdot \frac{1}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+199}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot t\_2\\ \end{array} \end{array} \]

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 (- (sqrt 2.0)))
        (t_3 (* (sqrt (* F (/ -0.5 C))) t_2)))
   (if (<= (pow B_m 2.0) 1e+33)
     (/ (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A))) t_1)
     (if (<= (pow B_m 2.0) 5e+139)
       t_3
       (if (<= (pow B_m 2.0) 1e+195)
         (* (* B_m (sqrt (* 2.0 (* F (- A (hypot B_m A)))))) (/ 1.0 t_1))
         (if (<= (pow B_m 2.0) 5e+199)
           t_3
           (* (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m))) t_2)))))))

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 = -sqrt(2.0);
	double t_3 = sqrt((F * (-0.5 / C))) * t_2;
	double tmp;
	if (pow(B_m, 2.0) <= 1e+33) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_1;
	} else if (pow(B_m, 2.0) <= 5e+139) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 1e+195) {
		tmp = (B_m * sqrt((2.0 * (F * (A - hypot(B_m, A)))))) * (1.0 / t_1);
	} else if (pow(B_m, 2.0) <= 5e+199) {
		tmp = t_3;
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_2;
	}
	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 = -Math.sqrt(2.0);
	double t_3 = Math.sqrt((F * (-0.5 / C))) * t_2;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e+33) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_1;
	} else if (Math.pow(B_m, 2.0) <= 5e+139) {
		tmp = t_3;
	} else if (Math.pow(B_m, 2.0) <= 1e+195) {
		tmp = (B_m * Math.sqrt((2.0 * (F * (A - Math.hypot(B_m, A)))))) * (1.0 / t_1);
	} else if (Math.pow(B_m, 2.0) <= 5e+199) {
		tmp = t_3;
	} else {
		tmp = Math.sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_2;
	}
	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 = -math.sqrt(2.0)
	t_3 = math.sqrt((F * (-0.5 / C))) * t_2
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e+33:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_1
	elif math.pow(B_m, 2.0) <= 5e+139:
		tmp = t_3
	elif math.pow(B_m, 2.0) <= 1e+195:
		tmp = (B_m * math.sqrt((2.0 * (F * (A - math.hypot(B_m, A)))))) * (1.0 / t_1)
	elif math.pow(B_m, 2.0) <= 5e+199:
		tmp = t_3
	else:
		tmp = math.sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_2
	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(-sqrt(2.0))
	t_3 = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_2)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+33)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / t_1);
	elseif ((B_m ^ 2.0) <= 5e+139)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 1e+195)
		tmp = Float64(Float64(B_m * sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A)))))) * Float64(1.0 / t_1));
	elseif ((B_m ^ 2.0) <= 5e+199)
		tmp = t_3;
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * t_2);
	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 = -sqrt(2.0);
	t_3 = sqrt((F * (-0.5 / C))) * t_2;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e+33)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / t_1;
	elseif ((B_m ^ 2.0) <= 5e+139)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 1e+195)
		tmp = (B_m * sqrt((2.0 * (F * (A - hypot(B_m, A)))))) * (1.0 / t_1);
	elseif ((B_m ^ 2.0) <= 5e+199)
		tmp = t_3;
	else
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_2;
	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[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$3 = N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+33], 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 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+139], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+195], N[(N[(B$95$m * N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+199], t$95$3, N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999995e32

   1. Initial program 27.8%

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

   3. Taylor expanded in A around -inf 21.5%

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

   if 9.9999999999999995e32 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e139 or 9.99999999999999977e194 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999998e199

   1. Initial program 19.7%

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

   3. Taylor expanded in F around 0 19.7%

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

   5. Simplified 35.4%

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

   6. Taylor expanded in A around -inf 25.5%

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

   if 5.0000000000000003e139 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999977e194

   1. Initial program 38.9%

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

   3. Taylor expanded in C around 0 38.9%

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

      1. associate-*l* 38.8%

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

      2. +-commutative 38.8%

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

      3. unpow2 38.8%

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

      4. unpow2 38.8%

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

      5. hypot-define 62.9%

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

   5. Simplified 62.9%

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

      1. div-inv 63.1%

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

      2. distribute-rgt-neg-in 63.1%

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

      3. sqrt-unprod 63.4%

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

      4. *-commutative 63.4%

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

      5. *-commutative 63.4%

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

   7. Applied egg-rr 63.4%

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

   if 4.9999999999999998e199 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 5.2%

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

   3. Taylor expanded in F around 0 8.6%

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

   5. Simplified 19.7%

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

   6. Taylor expanded in B around inf 29.9%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{\left(\frac{A}{B} + \frac{C}{B}\right) - 1}{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+33}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{+195}:\\ \;\;\;\;\left(B \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}\right) \cdot \frac{1}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-123}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{C \cdot \left(-4 \cdot \left(-A\right)\right) - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-23}:\\ \;\;\;\;-\sqrt{2 \cdot \left(-\frac{F}{B\_m}\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+250}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

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 2.0))))
   (if (<= (pow B_m 2.0) 1e-123)
     (/
      (sqrt (* -8.0 (* (* A C) (* F (+ A A)))))
      (- (* C (* -4.0 (- A))) (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 1e-23)
       (- (sqrt (* 2.0 (- (/ F B_m)))))
       (if (<= (pow B_m 2.0) 5e+250)
         (* (sqrt (* F (/ -0.5 C))) t_0)
         (* (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m))) t_0))))))

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(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 1e-123) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / ((C * (-4.0 * -A)) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 1e-23) {
		tmp = -sqrt((2.0 * -(F / B_m)));
	} else if (pow(B_m, 2.0) <= 5e+250) {
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_0;
	}
	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(2.0d0)
    if ((b_m ** 2.0d0) <= 1d-123) then
        tmp = sqrt(((-8.0d0) * ((a * c) * (f * (a + a))))) / ((c * ((-4.0d0) * -a)) - (b_m ** 2.0d0))
    else if ((b_m ** 2.0d0) <= 1d-23) then
        tmp = -sqrt((2.0d0 * -(f / b_m)))
    else if ((b_m ** 2.0d0) <= 5d+250) then
        tmp = sqrt((f * ((-0.5d0) / c))) * t_0
    else
        tmp = sqrt((f * ((((a / b_m) + (c / b_m)) + (-1.0d0)) / b_m))) * t_0
    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(2.0);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-123) {
		tmp = Math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / ((C * (-4.0 * -A)) - Math.pow(B_m, 2.0));
	} else if (Math.pow(B_m, 2.0) <= 1e-23) {
		tmp = -Math.sqrt((2.0 * -(F / B_m)));
	} else if (Math.pow(B_m, 2.0) <= 5e+250) {
		tmp = Math.sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = Math.sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_0;
	}
	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(2.0)
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-123:
		tmp = math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / ((C * (-4.0 * -A)) - math.pow(B_m, 2.0))
	elif math.pow(B_m, 2.0) <= 1e-23:
		tmp = -math.sqrt((2.0 * -(F / B_m)))
	elif math.pow(B_m, 2.0) <= 5e+250:
		tmp = math.sqrt((F * (-0.5 / C))) * t_0
	else:
		tmp = math.sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_0
	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(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-123)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(Float64(C * Float64(-4.0 * Float64(-A))) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 1e-23)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(-Float64(F / B_m)))));
	elseif ((B_m ^ 2.0) <= 5e+250)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * t_0);
	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(2.0);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-123)
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / ((C * (-4.0 * -A)) - (B_m ^ 2.0));
	elseif ((B_m ^ 2.0) <= 1e-23)
		tmp = -sqrt((2.0 * -(F / B_m)));
	elseif ((B_m ^ 2.0) <= 5e+250)
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	else
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_0;
	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[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-123], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(C * N[(-4.0 * (-A)), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-23], (-N[Sqrt[N[(2.0 * (-N[(F / B$95$m), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+250], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.0000000000000001e-123

   1. Initial program 25.4%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in C around inf 20.1%

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

      1. associate-*r* 20.9%

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

      2. *-commutative 20.9%

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

      3. mul-1-neg 20.9%

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

   7. Simplified 20.9%

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

      1. fma-undefine 20.9%

         \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{\left(C \cdot \left(A \cdot -4\right) + {B}^{2}\right)}} \]

   9. Applied egg-rr 20.9%

      \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{\left(C \cdot \left(A \cdot -4\right) + {B}^{2}\right)}} \]

   if 1.0000000000000001e-123 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999996e-24

   1. Initial program 38.2%

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

   3. Taylor expanded in F around 0 38.1%

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

   5. Simplified 53.0%

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

   6. Taylor expanded in B around inf 13.4%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-1}{B}}} \]
   7. Step-by-step derivation

      1. pow1 13.4%

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

      2. sqrt-unprod 13.4%

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

   8. Applied egg-rr 13.4%

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

      1. unpow1 13.4%

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

      2. associate-*r/ 13.5%

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

      3. *-commutative 13.5%

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

      4. mul-1-neg 13.5%

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

   10. Simplified 13.5%

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

   if 9.9999999999999996e-24 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000002e250

   1. Initial program 27.5%

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

   3. Taylor expanded in F around 0 24.2%

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

   5. Simplified 40.6%

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

   6. Taylor expanded in A around -inf 22.4%

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

   if 5.0000000000000002e250 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.8%

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

   3. Taylor expanded in F around 0 8.1%

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

   5. Simplified 17.9%

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

   6. Taylor expanded in B around inf 33.0%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{\left(\frac{A}{B} + \frac{C}{B}\right) - 1}{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-123}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{C \cdot \left(-4 \cdot \left(-A\right)\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-23}:\\ \;\;\;\;-\sqrt{2 \cdot \left(-\frac{F}{B}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+250}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 40.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 6 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{C \cdot \left(-4 \cdot \left(-A\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-23}:\\ \;\;\;\;-\sqrt{2 \cdot \left(-\frac{F}{B\_m}\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

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 2.0))))
   (if (<= (pow B_m 2.0) 6e-124)
     (/ (sqrt (* -8.0 (* (* A C) (* F (+ A A))))) (* C (* -4.0 (- A))))
     (if (<= (pow B_m 2.0) 1e-23)
       (- (sqrt (* 2.0 (- (/ F B_m)))))
       (if (<= (pow B_m 2.0) 5e+139)
         (* (sqrt (* F (/ -0.5 C))) t_0)
         (* (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m))) t_0))))))

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(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 6e-124) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (C * (-4.0 * -A));
	} else if (pow(B_m, 2.0) <= 1e-23) {
		tmp = -sqrt((2.0 * -(F / B_m)));
	} else if (pow(B_m, 2.0) <= 5e+139) {
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_0;
	}
	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(2.0d0)
    if ((b_m ** 2.0d0) <= 6d-124) then
        tmp = sqrt(((-8.0d0) * ((a * c) * (f * (a + a))))) / (c * ((-4.0d0) * -a))
    else if ((b_m ** 2.0d0) <= 1d-23) then
        tmp = -sqrt((2.0d0 * -(f / b_m)))
    else if ((b_m ** 2.0d0) <= 5d+139) then
        tmp = sqrt((f * ((-0.5d0) / c))) * t_0
    else
        tmp = sqrt((f * ((((a / b_m) + (c / b_m)) + (-1.0d0)) / b_m))) * t_0
    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(2.0);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 6e-124) {
		tmp = Math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (C * (-4.0 * -A));
	} else if (Math.pow(B_m, 2.0) <= 1e-23) {
		tmp = -Math.sqrt((2.0 * -(F / B_m)));
	} else if (Math.pow(B_m, 2.0) <= 5e+139) {
		tmp = Math.sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = Math.sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_0;
	}
	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(2.0)
	tmp = 0
	if math.pow(B_m, 2.0) <= 6e-124:
		tmp = math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (C * (-4.0 * -A))
	elif math.pow(B_m, 2.0) <= 1e-23:
		tmp = -math.sqrt((2.0 * -(F / B_m)))
	elif math.pow(B_m, 2.0) <= 5e+139:
		tmp = math.sqrt((F * (-0.5 / C))) * t_0
	else:
		tmp = math.sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_0
	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(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 6e-124)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(C * Float64(-4.0 * Float64(-A))));
	elseif ((B_m ^ 2.0) <= 1e-23)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(-Float64(F / B_m)))));
	elseif ((B_m ^ 2.0) <= 5e+139)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m))) * t_0);
	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(2.0);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 6e-124)
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (C * (-4.0 * -A));
	elseif ((B_m ^ 2.0) <= 1e-23)
		tmp = -sqrt((2.0 * -(F / B_m)));
	elseif ((B_m ^ 2.0) <= 5e+139)
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	else
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m))) * t_0;
	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[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 6e-124], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(-4.0 * (-A)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-23], (-N[Sqrt[N[(2.0 * (-N[(F / B$95$m), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+139], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 6e-124

   1. Initial program 24.9%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in C around inf 20.4%

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

      1. associate-*r* 21.3%

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

      2. *-commutative 21.3%

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

      3. mul-1-neg 21.3%

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

   7. Simplified 21.3%

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

      1. add-log-exp 1.5%

         \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{\log \left(e^{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\right)}} \]

   9. Applied egg-rr 1.5%

      \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{\log \left(e^{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\right)}} \]

   10. Taylor expanded in C around inf 21.1%

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

       1. *-commutative 21.1%

          \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{\left(A \cdot C\right) \cdot -4}} \]

       2. *-commutative 21.1%

          \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{\left(C \cdot A\right)} \cdot -4} \]

       3. associate-*r* 21.1%

          \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{C \cdot \left(A \cdot -4\right)}} \]

   12. Simplified 21.1%

       \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{C \cdot \left(A \cdot -4\right)}} \]

   if 6e-124 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999996e-24

   1. Initial program 39.7%

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

   3. Taylor expanded in F around 0 34.0%

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

   5. Simplified 52.6%

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

   6. Taylor expanded in B around inf 12.6%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-1}{B}}} \]
   7. Step-by-step derivation

      1. pow1 12.6%

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

      2. sqrt-unprod 12.6%

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

   8. Applied egg-rr 12.6%

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

      1. unpow1 12.6%

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

      2. associate-*r/ 12.7%

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

      3. *-commutative 12.7%

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

      4. mul-1-neg 12.7%

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

   10. Simplified 12.7%

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

   if 9.9999999999999996e-24 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e139

   1. Initial program 25.4%

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

   3. Taylor expanded in F around 0 25.3%

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

   5. Simplified 37.1%

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

   6. Taylor expanded in A around -inf 22.3%

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

   if 5.0000000000000003e139 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 7.9%

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

   3. Taylor expanded in F around 0 11.0%

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

   5. Simplified 24.0%

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

   6. Taylor expanded in B around inf 29.4%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{\left(\frac{A}{B} + \frac{C}{B}\right) - 1}{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 23.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 6 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{C \cdot \left(-4 \cdot \left(-A\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-23}:\\ \;\;\;\;-\sqrt{2 \cdot \left(-\frac{F}{B}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 40.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 6 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{C \cdot \left(-4 \cdot \left(-A\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-23} \lor \neg \left({B\_m}^{2} \leq 5 \cdot 10^{+139}\right):\\ \;\;\;\;-\sqrt{2 \cdot \left(-\frac{F}{B\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 6e-124)
   (/ (sqrt (* -8.0 (* (* A C) (* F (+ A A))))) (* C (* -4.0 (- A))))
   (if (or (<= (pow B_m 2.0) 1e-23) (not (<= (pow B_m 2.0) 5e+139)))
     (- (sqrt (* 2.0 (- (/ F B_m)))))
     (* (sqrt (* F (/ -0.5 C))) (- (sqrt 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 6e-124) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (C * (-4.0 * -A));
	} else if ((pow(B_m, 2.0) <= 1e-23) || !(pow(B_m, 2.0) <= 5e+139)) {
		tmp = -sqrt((2.0 * -(F / B_m)));
	} else {
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if ((b_m ** 2.0d0) <= 6d-124) then
        tmp = sqrt(((-8.0d0) * ((a * c) * (f * (a + a))))) / (c * ((-4.0d0) * -a))
    else if (((b_m ** 2.0d0) <= 1d-23) .or. (.not. ((b_m ** 2.0d0) <= 5d+139))) then
        tmp = -sqrt((2.0d0 * -(f / b_m)))
    else
        tmp = sqrt((f * ((-0.5d0) / c))) * -sqrt(2.0d0)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 6e-124) {
		tmp = Math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (C * (-4.0 * -A));
	} else if ((Math.pow(B_m, 2.0) <= 1e-23) || !(Math.pow(B_m, 2.0) <= 5e+139)) {
		tmp = -Math.sqrt((2.0 * -(F / B_m)));
	} else {
		tmp = Math.sqrt((F * (-0.5 / C))) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 6e-124:
		tmp = math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (C * (-4.0 * -A))
	elif (math.pow(B_m, 2.0) <= 1e-23) or not (math.pow(B_m, 2.0) <= 5e+139):
		tmp = -math.sqrt((2.0 * -(F / B_m)))
	else:
		tmp = math.sqrt((F * (-0.5 / C))) * -math.sqrt(2.0)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 6e-124)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(C * Float64(-4.0 * Float64(-A))));
	elseif (((B_m ^ 2.0) <= 1e-23) || !((B_m ^ 2.0) <= 5e+139))
		tmp = Float64(-sqrt(Float64(2.0 * Float64(-Float64(F / B_m)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 6e-124)
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (C * (-4.0 * -A));
	elseif (((B_m ^ 2.0) <= 1e-23) || ~(((B_m ^ 2.0) <= 5e+139)))
		tmp = -sqrt((2.0 * -(F / B_m)));
	else
		tmp = sqrt((F * (-0.5 / C))) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 6e-124], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(-4.0 * (-A)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-23], N[Not[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+139]], $MachinePrecision]], (-N[Sqrt[N[(2.0 * (-N[(F / B$95$m), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 6e-124

   1. Initial program 24.9%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in C around inf 20.4%

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

      1. associate-*r* 21.3%

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

      2. *-commutative 21.3%

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

      3. mul-1-neg 21.3%

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

   7. Simplified 21.3%

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

      1. add-log-exp 1.5%

         \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{\log \left(e^{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\right)}} \]

   9. Applied egg-rr 1.5%

      \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{\log \left(e^{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\right)}} \]

   10. Taylor expanded in C around inf 21.1%

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

       1. *-commutative 21.1%

          \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{\left(A \cdot C\right) \cdot -4}} \]

       2. *-commutative 21.1%

          \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{\left(C \cdot A\right)} \cdot -4} \]

       3. associate-*r* 21.1%

          \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{C \cdot \left(A \cdot -4\right)}} \]

   12. Simplified 21.1%

       \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{C \cdot \left(A \cdot -4\right)}} \]

   if 6e-124 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999996e-24 or 5.0000000000000003e139 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.9%

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

   3. Taylor expanded in F around 0 14.6%

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

   5. Simplified 28.5%

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

   6. Taylor expanded in B around inf 25.5%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-1}{B}}} \]
   7. Step-by-step derivation

      1. pow1 25.5%

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

      2. sqrt-unprod 25.6%

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

   8. Applied egg-rr 25.6%

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

      1. unpow1 25.6%

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

      2. associate-*r/ 25.6%

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

      3. *-commutative 25.6%

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

      4. mul-1-neg 25.6%

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

   10. Simplified 25.6%

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

   if 9.9999999999999996e-24 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e139

   1. Initial program 25.4%

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

   3. Taylor expanded in F around 0 25.3%

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

   5. Simplified 37.1%

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

   6. Taylor expanded in A around -inf 22.3%

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

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 6 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{C \cdot \left(-4 \cdot \left(-A\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-23} \lor \neg \left({B}^{2} \leq 5 \cdot 10^{+139}\right):\\ \;\;\;\;-\sqrt{2 \cdot \left(-\frac{F}{B}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 41.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.4 \cdot 10^{-62} \lor \neg \left(B\_m \leq 1.06 \cdot 10^{-10}\right) \land B\_m \leq 0.0022:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{C \cdot \left(-4 \cdot \left(-A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \left(-\frac{F}{B\_m}\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (or (<= B_m 2.4e-62) (and (not (<= B_m 1.06e-10)) (<= B_m 0.0022)))
   (/ (sqrt (* -8.0 (* (* A C) (* F (+ A A))))) (* C (* -4.0 (- A))))
   (- (sqrt (* 2.0 (- (/ F B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if ((B_m <= 2.4e-62) || (!(B_m <= 1.06e-10) && (B_m <= 0.0022))) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (C * (-4.0 * -A));
	} else {
		tmp = -sqrt((2.0 * -(F / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if ((b_m <= 2.4d-62) .or. (.not. (b_m <= 1.06d-10)) .and. (b_m <= 0.0022d0)) then
        tmp = sqrt(((-8.0d0) * ((a * c) * (f * (a + a))))) / (c * ((-4.0d0) * -a))
    else
        tmp = -sqrt((2.0d0 * -(f / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if ((B_m <= 2.4e-62) || (!(B_m <= 1.06e-10) && (B_m <= 0.0022))) {
		tmp = Math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (C * (-4.0 * -A));
	} else {
		tmp = -Math.sqrt((2.0 * -(F / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if (B_m <= 2.4e-62) or (not (B_m <= 1.06e-10) and (B_m <= 0.0022)):
		tmp = math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (C * (-4.0 * -A))
	else:
		tmp = -math.sqrt((2.0 * -(F / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m <= 2.4e-62) || (!(B_m <= 1.06e-10) && (B_m <= 0.0022)))
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(C * Float64(-4.0 * Float64(-A))));
	else
		tmp = Float64(-sqrt(Float64(2.0 * Float64(-Float64(F / B_m)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m <= 2.4e-62) || (~((B_m <= 1.06e-10)) && (B_m <= 0.0022)))
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (C * (-4.0 * -A));
	else
		tmp = -sqrt((2.0 * -(F / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[Or[LessEqual[B$95$m, 2.4e-62], And[N[Not[LessEqual[B$95$m, 1.06e-10]], $MachinePrecision], LessEqual[B$95$m, 0.0022]]], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(-4.0 * (-A)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(2.0 * (-N[(F / B$95$m), $MachinePrecision])), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 2.39999999999999984e-62 or 1.06e-10 < B < 0.00220000000000000013

   1. Initial program 21.2%

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

   3. Simplified 23.8%

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

   5. Taylor expanded in C around inf 14.0%

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

      1. associate-*r* 14.4%

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

      2. *-commutative 14.4%

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

      3. mul-1-neg 14.4%

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

   7. Simplified 14.4%

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

      1. add-log-exp 1.7%

         \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{\log \left(e^{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\right)}} \]

   9. Applied egg-rr 1.7%

      \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{\log \left(e^{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\right)}} \]

   10. Taylor expanded in C around inf 15.5%

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

       1. *-commutative 15.5%

          \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{\left(A \cdot C\right) \cdot -4}} \]

       2. *-commutative 15.5%

          \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{\left(C \cdot A\right)} \cdot -4} \]

       3. associate-*r* 15.5%

          \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{C \cdot \left(A \cdot -4\right)}} \]

   12. Simplified 15.5%

       \[\leadsto \frac{\sqrt{-8 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}}{-\color{blue}{C \cdot \left(A \cdot -4\right)}} \]

   if 2.39999999999999984e-62 < B < 1.06e-10 or 0.00220000000000000013 < B

   1. Initial program 15.7%

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

   3. Taylor expanded in F around 0 15.6%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in B around inf 42.4%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-1}{B}}} \]
   7. Step-by-step derivation

      1. pow1 42.4%

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

      2. sqrt-unprod 42.5%

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

   8. Applied egg-rr 42.5%

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

      1. unpow1 42.5%

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

      2. associate-*r/ 42.6%

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

      3. *-commutative 42.6%

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

      4. mul-1-neg 42.6%

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

   10. Simplified 42.6%

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

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.4 \cdot 10^{-62} \lor \neg \left(B \leq 1.06 \cdot 10^{-10}\right) \land B \leq 0.0022:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{C \cdot \left(-4 \cdot \left(-A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \left(-\frac{F}{B}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.9% accurate, 5.9× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* 2.0 (- (/ 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(-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 19.6%

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

3. Taylor expanded in F around 0 17.7%

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

5. Simplified 28.1%

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

6. Taylor expanded in B around inf 15.3%

   \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-1}{B}}} \]
7. Step-by-step derivation

   1. pow1 15.3%

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

   2. sqrt-unprod 15.3%

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

8. Applied egg-rr 15.3%

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

   1. unpow1 15.3%

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

   2. associate-*r/ 15.3%

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

   3. *-commutative 15.3%

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

   4. mul-1-neg 15.3%

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

10. Simplified 15.3%

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

11. Final simplification 15.3%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :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))))