ABCF->ab-angle b

Percentage Accurate: 19.0% → 48.9%

Time: 32.7s

Alternatives: 9

Speedup: 5.3×

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: 48.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := t\_1 - {B\_m}^{2}\\ t_3 := {B\_m}^{2} - t\_1\\ t_4 := 2 \cdot \left(t\_3 \cdot F\right)\\ t_5 := \frac{\sqrt{t\_4 \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_5 \leq -1 \cdot 10^{-188}:\\ \;\;\;\;\frac{-1}{\frac{t\_3}{{\left(2 \cdot t\_3\right)}^{0.5} \cdot \sqrt{F \cdot \left(\left(A + C\right) - t\_0\right)}}}\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{\sqrt{t\_4 \cdot \left(A + \left(A + -0.5 \cdot \frac{\left({B\_m}^{2} + {A}^{2}\right) - {A}^{2}}{C}\right)\right)}}{t\_2}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot F} \cdot \sqrt{\left(C - \left(t\_0 - A\right)\right) \cdot \mathsf{fma}\left(A, C \cdot -4, {B\_m}^{2}\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{B\_m}\right)\right)\\ \end{array} \end{array} \]

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 (hypot B_m (- A C)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- t_1 (pow B_m 2.0)))
        (t_3 (- (pow B_m 2.0) t_1))
        (t_4 (* 2.0 (* t_3 F)))
        (t_5
         (/
          (sqrt (* t_4 (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2)))
   (if (<= t_5 -1e-188)
     (/ -1.0 (/ t_3 (* (pow (* 2.0 t_3) 0.5) (sqrt (* F (- (+ A C) t_0))))))
     (if (<= t_5 5e+115)
       (/
        (sqrt
         (*
          t_4
          (+
           A
           (+
            A
            (* -0.5 (/ (- (+ (pow B_m 2.0) (pow A 2.0)) (pow A 2.0)) C))))))
        t_2)
       (if (<= t_5 INFINITY)
         (/
          (*
           (sqrt (* 2.0 F))
           (sqrt (* (- C (- t_0 A)) (fma A (* C -4.0) (pow B_m 2.0)))))
          (- (* 4.0 (* A C)) (pow B_m 2.0)))
         (-
          (expm1
           (log1p (/ (sqrt (* 2.0 (* F (- A (hypot B_m A))))) 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 = hypot(B_m, (A - C));
	double t_1 = (4.0 * A) * C;
	double t_2 = t_1 - pow(B_m, 2.0);
	double t_3 = pow(B_m, 2.0) - t_1;
	double t_4 = 2.0 * (t_3 * F);
	double t_5 = sqrt((t_4 * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double tmp;
	if (t_5 <= -1e-188) {
		tmp = -1.0 / (t_3 / (pow((2.0 * t_3), 0.5) * sqrt((F * ((A + C) - t_0)))));
	} else if (t_5 <= 5e+115) {
		tmp = sqrt((t_4 * (A + (A + (-0.5 * (((pow(B_m, 2.0) + pow(A, 2.0)) - pow(A, 2.0)) / C)))))) / t_2;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * F)) * sqrt(((C - (t_0 - A)) * fma(A, (C * -4.0), pow(B_m, 2.0))))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else {
		tmp = -expm1(log1p((sqrt((2.0 * (F * (A - hypot(B_m, A))))) / 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 = hypot(B_m, Float64(A - C))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	t_3 = Float64((B_m ^ 2.0) - t_1)
	t_4 = Float64(2.0 * Float64(t_3 * F))
	t_5 = Float64(sqrt(Float64(t_4 * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	tmp = 0.0
	if (t_5 <= -1e-188)
		tmp = Float64(-1.0 / Float64(t_3 / Float64((Float64(2.0 * t_3) ^ 0.5) * sqrt(Float64(F * Float64(Float64(A + C) - t_0))))));
	elseif (t_5 <= 5e+115)
		tmp = Float64(sqrt(Float64(t_4 * Float64(A + Float64(A + Float64(-0.5 * Float64(Float64(Float64((B_m ^ 2.0) + (A ^ 2.0)) - (A ^ 2.0)) / C)))))) / t_2);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * F)) * sqrt(Float64(Float64(C - Float64(t_0 - A)) * fma(A, Float64(C * -4.0), (B_m ^ 2.0))))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	else
		tmp = Float64(-expm1(log1p(Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))) / 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[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t$95$4 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$5, -1e-188], N[(-1.0 / N[(t$95$3 / N[(N[Power[N[(2.0 * t$95$3), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A + C), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 5e+115], N[(N[Sqrt[N[(t$95$4 * N[(A + N[(A + N[(-0.5 * N[(N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[A, 2.0], $MachinePrecision]), $MachinePrecision] - N[Power[A, 2.0], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(C - N[(t$95$0 - A), $MachinePrecision]), $MachinePrecision] * N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(Exp[N[Log[1 + N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision])]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -9.9999999999999995e-189

   1. Initial program 38.8%

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

   4. Applied egg-rr 50.7%

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

      1. unpow-1 50.7%

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

      2. distribute-frac-neg2 50.7%

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

      3. associate-*l* 50.5%

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

      4. hypot-undefine 38.7%

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

      5. unpow2 38.7%

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

      6. unpow2 38.7%

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

      7. +-commutative 38.7%

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

      8. unpow2 38.7%

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

   6. Simplified 50.5%

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

      1. pow1/2 50.5%

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

      2. associate-*r* 50.5%

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

      3. *-commutative 50.5%

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

      4. *-commutative 50.5%

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

      5. unpow-prod-down 71.6%

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

      6. *-commutative 71.6%

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

      7. *-commutative 71.6%

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

      8. pow1/2 71.6%

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

      9. associate-+r- 71.0%

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

   8. Applied egg-rr 71.0%

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

   if -9.9999999999999995e-189 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 5.00000000000000008e115

   1. Initial program 17.1%

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

   3. Taylor expanded in C around inf 33.7%

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

      1. associate--l+ 33.7%

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

      2. +-commutative 33.7%

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

      3. unpow2 33.7%

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

      4. mul-1-neg 33.7%

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

      5. mul-1-neg 33.7%

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

      6. sqr-neg 33.7%

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

      7. unpow2 33.7%

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

      8. mul-1-neg 33.7%

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

   5. Simplified 33.7%

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

   if 5.00000000000000008e115 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 26.0%

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

   3. Simplified 40.8%

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

      1. pow1/2 40.8%

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

      2. associate-*r* 40.8%

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

      3. unpow-prod-down 55.2%

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

      4. pow1/2 55.2%

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

      5. associate-+r- 55.2%

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

      6. +-commutative 55.2%

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

      7. associate-+r- 55.2%

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

      8. hypot-undefine 26.1%

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

      9. unpow2 26.1%

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

      10. unpow2 26.1%

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

      11. +-commutative 26.1%

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

      12. unpow2 26.1%

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

      13. unpow2 26.1%

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

      14. hypot-define 55.2%

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

   6. Applied egg-rr 55.2%

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

      1. unpow1/2 55.2%

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

      2. *-commutative 55.2%

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

      3. hypot-undefine 26.1%

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

      4. unpow2 26.1%

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

      5. unpow2 26.1%

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

      6. +-commutative 26.1%

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

      7. unpow2 26.1%

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

      8. unpow2 26.1%

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

      9. hypot-undefine 55.2%

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

   8. Simplified 55.2%

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 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 C around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. +-commutative 1.7%

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

      3. unpow2 1.7%

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

      4. unpow2 1.7%

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

      5. hypot-define 18.7%

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

   5. Simplified 18.7%

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

   7. Applied egg-rr 3.8%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)\right)}^{0.5}}{B}\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-define 18.7%

         \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)\right)}^{0.5}}{B}\right)\right)} \]

      2. unpow1/2 18.7%

         \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}}{B}\right)\right) \]

      3. hypot-undefine 1.2%

         \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \color{blue}{\sqrt{A \cdot A + B \cdot B}}\right)\right)}}{B}\right)\right) \]

      4. unpow2 1.2%

         \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\color{blue}{{A}^{2}} + B \cdot B}\right)\right)}}{B}\right)\right) \]

      5. unpow2 1.2%

         \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{{A}^{2} + \color{blue}{{B}^{2}}}\right)\right)}}{B}\right)\right) \]

      6. +-commutative 1.2%

         \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\color{blue}{{B}^{2} + {A}^{2}}}\right)\right)}}{B}\right)\right) \]

      7. unpow2 1.2%

         \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)\right)}}{B}\right)\right) \]

      8. unpow2 1.2%

         \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right)\right)}}{B}\right)\right) \]

      9. hypot-undefine 18.7%

         \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \color{blue}{\mathsf{hypot}\left(B, A\right)}\right)\right)}}{B}\right)\right) \]

   9. Simplified 18.7%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-188}:\\ \;\;\;\;\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}^{0.5} \cdot \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {A}^{2}}{C}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot F} \cdot \sqrt{\left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right) \cdot \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 46.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := {B\_m}^{2} - t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-192}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+297}:\\ \;\;\;\;\frac{-1}{\frac{t\_1}{{\left(2 \cdot t\_1\right)}^{0.5} \cdot \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)) (t_1 (- (pow B_m 2.0) t_0)))
   (if (<= (pow B_m 2.0) 1e-192)
     (/ (sqrt (* (* 2.0 (* t_1 F)) (* 2.0 A))) (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 4e+297)
       (/
        -1.0
        (/
         t_1
         (*
          (pow (* 2.0 t_1) 0.5)
          (sqrt (* F (- (+ A C) (hypot B_m (- A C))))))))
       (/ (sqrt (* 2.0 (* F (- A (hypot B_m A))))) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = pow(B_m, 2.0) - t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-192) {
		tmp = sqrt(((2.0 * (t_1 * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 4e+297) {
		tmp = -1.0 / (t_1 / (pow((2.0 * t_1), 0.5) * sqrt((F * ((A + C) - hypot(B_m, (A - C)))))));
	} else {
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = Math.pow(B_m, 2.0) - t_0;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-192) {
		tmp = Math.sqrt(((2.0 * (t_1 * F)) * (2.0 * A))) / (t_0 - Math.pow(B_m, 2.0));
	} else if (Math.pow(B_m, 2.0) <= 4e+297) {
		tmp = -1.0 / (t_1 / (Math.pow((2.0 * t_1), 0.5) * Math.sqrt((F * ((A + C) - Math.hypot(B_m, (A - C)))))));
	} else {
		tmp = Math.sqrt((2.0 * (F * (A - Math.hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	t_1 = math.pow(B_m, 2.0) - t_0
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-192:
		tmp = math.sqrt(((2.0 * (t_1 * F)) * (2.0 * A))) / (t_0 - math.pow(B_m, 2.0))
	elif math.pow(B_m, 2.0) <= 4e+297:
		tmp = -1.0 / (t_1 / (math.pow((2.0 * t_1), 0.5) * math.sqrt((F * ((A + C) - math.hypot(B_m, (A - C)))))))
	else:
		tmp = math.sqrt((2.0 * (F * (A - math.hypot(B_m, A))))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64((B_m ^ 2.0) - t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-192)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 4e+297)
		tmp = Float64(-1.0 / Float64(t_1 / Float64((Float64(2.0 * t_1) ^ 0.5) * sqrt(Float64(F * Float64(Float64(A + C) - hypot(B_m, Float64(A - C))))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	t_1 = (B_m ^ 2.0) - t_0;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-192)
		tmp = sqrt(((2.0 * (t_1 * F)) * (2.0 * A))) / (t_0 - (B_m ^ 2.0));
	elseif ((B_m ^ 2.0) <= 4e+297)
		tmp = -1.0 / (t_1 / (((2.0 * t_1) ^ 0.5) * sqrt((F * ((A + C) - hypot(B_m, (A - C)))))));
	else
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-192], N[(N[Sqrt[N[(N[(2.0 * N[(t$95$1 * 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], 4e+297], N[(-1.0 / N[(t$95$1 / N[(N[Power[N[(2.0 * t$95$1), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 1.0000000000000001e-192

   1. Initial program 18.8%

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

   3. Taylor expanded in A around -inf 21.8%

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

   if 1.0000000000000001e-192 < (pow.f64 B 2) < 4.0000000000000001e297

   1. Initial program 31.2%

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

   4. Applied egg-rr 40.0%

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

      1. unpow-1 40.0%

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

      2. distribute-frac-neg2 40.0%

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

      3. associate-*l* 40.9%

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

      4. hypot-undefine 32.2%

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

      5. unpow2 32.2%

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

      6. unpow2 32.2%

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

      7. +-commutative 32.2%

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

      8. unpow2 32.2%

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

   6. Simplified 40.9%

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

      1. pow1/2 40.9%

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

      2. associate-*r* 40.9%

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

      3. *-commutative 40.9%

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

      4. *-commutative 40.9%

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

      5. unpow-prod-down 57.7%

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

      6. *-commutative 57.7%

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

      7. *-commutative 57.7%

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

      8. pow1/2 57.7%

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

      9. associate-+r- 57.3%

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

   8. Applied egg-rr 57.3%

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

   if 4.0000000000000001e297 < (pow.f64 B 2)

   1. Initial program 1.7%

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

   3. Taylor expanded in C around 0 3.0%

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

      1. mul-1-neg 3.0%

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

      2. +-commutative 3.0%

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

      3. unpow2 3.0%

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

      4. unpow2 3.0%

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

      5. hypot-define 29.1%

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

   5. Simplified 29.1%

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

      1. associate-*l/ 29.2%

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

   7. Applied egg-rr 29.3%

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

      1. unpow1/2 29.2%

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

      2. hypot-undefine 3.0%

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

      3. unpow2 3.0%

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

      4. unpow2 3.0%

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

      5. +-commutative 3.0%

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

      6. unpow2 3.0%

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

      7. unpow2 3.0%

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

      8. hypot-undefine 29.2%

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

   9. Simplified 29.2%

      \[\leadsto -\color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-192}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+297}:\\ \;\;\;\;\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}^{0.5} \cdot \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+18}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(F \cdot \left(2 \cdot \left(C + \left(A - \mathsf{hypot}\left(A - C, B\_m\right)\right)\right)\right)\right)} \cdot \frac{-1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)) (t_1 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 4e-159)
     (/
      (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+18)
       (*
        (sqrt (* t_1 (* F (* 2.0 (+ C (- A (hypot (- A C) B_m)))))))
        (/ -1.0 t_1))
       (/ (sqrt (* 2.0 (* F (- A (hypot B_m A))))) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 4e-159) {
		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+18) {
		tmp = sqrt((t_1 * (F * (2.0 * (C + (A - hypot((A - C), B_m))))))) * (-1.0 / t_1);
	} else {
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-159)
		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+18)
		tmp = Float64(sqrt(Float64(t_1 * Float64(F * Float64(2.0 * Float64(C + Float64(A - hypot(Float64(A - C), B_m))))))) * Float64(-1.0 / t_1));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-159], 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+18], N[(N[Sqrt[N[(t$95$1 * N[(F * N[(2.0 * N[(C + N[(A - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 3.99999999999999995e-159

   1. Initial program 18.4%

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

   3. Taylor expanded in A around -inf 22.5%

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

   if 3.99999999999999995e-159 < (pow.f64 B 2) < 1e18

   1. Initial program 45.4%

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

   3. Simplified 56.1%

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

      1. div-inv 56.0%

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

   6. Applied egg-rr 58.6%

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

   if 1e18 < (pow.f64 B 2)

   1. Initial program 13.5%

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

   3. Taylor expanded in C around 0 11.9%

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

      1. mul-1-neg 11.9%

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

      2. +-commutative 11.9%

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

      3. unpow2 11.9%

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

      4. unpow2 11.9%

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

      5. hypot-define 26.7%

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

   5. Simplified 26.7%

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

      1. associate-*l/ 26.8%

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

   7. Applied egg-rr 26.9%

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

      1. unpow1/2 26.9%

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

      2. hypot-undefine 11.9%

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

      3. unpow2 11.9%

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

      4. unpow2 11.9%

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

      5. +-commutative 11.9%

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

      6. unpow2 11.9%

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

      7. unpow2 11.9%

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

      8. hypot-undefine 26.9%

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

   9. Simplified 26.9%

      \[\leadsto -\color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+18}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(C + \left(A - \mathsf{hypot}\left(A - C, B\right)\right)\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 2e-25)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (/ (sqrt (* 2.0 (* F (- A (hypot B_m A))))) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 2e-25) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-25) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt((2.0 * (F * (A - Math.hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-25:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt((2.0 * (F * (A - math.hypot(B_m, A))))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-25)
		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)));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-25)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-25], 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], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 2.00000000000000008e-25

   1. Initial program 22.9%

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

   3. Taylor expanded in A around -inf 21.4%

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

   if 2.00000000000000008e-25 < (pow.f64 B 2)

   1. Initial program 17.5%

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

   3. Taylor expanded in C around 0 12.6%

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

      1. mul-1-neg 12.6%

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

      2. +-commutative 12.6%

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

      3. unpow2 12.6%

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

      4. unpow2 12.6%

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

      5. hypot-define 26.3%

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

   5. Simplified 26.3%

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

      1. associate-*l/ 26.3%

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

   7. Applied egg-rr 26.4%

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

      1. unpow1/2 26.4%

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

      2. hypot-undefine 12.7%

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

      3. unpow2 12.7%

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

      4. unpow2 12.7%

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

      5. +-commutative 12.7%

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

      6. unpow2 12.7%

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

      7. unpow2 12.7%

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

      8. hypot-undefine 26.4%

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

   9. Simplified 26.4%

      \[\leadsto -\color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.1%

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

Alternative 5: 44.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-146}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-146)
   (/ (sqrt (* 2.0 (* -4.0 (* A (* C (* F (+ A A))))))) (* 4.0 (* A C)))
   (/ (sqrt (* 2.0 (* F (- A (hypot B_m A))))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-146) {
		tmp = sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C));
	} else {
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-146) {
		tmp = Math.sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C));
	} else {
		tmp = Math.sqrt((2.0 * (F * (A - Math.hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-146:
		tmp = math.sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C))
	else:
		tmp = math.sqrt((2.0 * (F * (A - math.hypot(B_m, A))))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-146)
		tmp = Float64(sqrt(Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * Float64(F * Float64(A + A))))))) / Float64(4.0 * Float64(A * C)));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-146)
		tmp = sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C));
	else
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-146], N[(N[Sqrt[N[(2.0 * N[(-4.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 2.00000000000000005e-146

   1. Initial program 18.3%

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

   3. Simplified 23.1%

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

   5. Taylor expanded in C around inf 20.1%

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

      1. mul-1-neg 20.1%

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

   7. Simplified 20.1%

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

   8. Taylor expanded in A around inf 20.0%

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

   if 2.00000000000000005e-146 < (pow.f64 B 2)

   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 C around 0 14.0%

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

      1. mul-1-neg 14.0%

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

      2. +-commutative 14.0%

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

      3. unpow2 14.0%

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

      4. unpow2 14.0%

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

      5. hypot-define 25.5%

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

   5. Simplified 25.5%

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

      1. associate-*l/ 25.6%

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

   7. Applied egg-rr 25.7%

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

      1. unpow1/2 25.7%

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

      2. hypot-undefine 14.0%

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

      3. unpow2 14.0%

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

      4. unpow2 14.0%

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

      5. +-commutative 14.0%

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

      6. unpow2 14.0%

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

      7. unpow2 14.0%

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

      8. hypot-undefine 25.7%

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

   9. Simplified 25.7%

      \[\leadsto -\color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-146}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.4 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\_m\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.4e-55)
   (/ (sqrt (* 2.0 (* -4.0 (* A (* C (* F (+ A A))))))) (* 4.0 (* A C)))
   (* (sqrt (* F (- A B_m))) (/ (sqrt 2.0) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.4e-55) {
		tmp = sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C));
	} else {
		tmp = sqrt((F * (A - B_m))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 4.4d-55) then
        tmp = sqrt((2.0d0 * ((-4.0d0) * (a * (c * (f * (a + a))))))) / (4.0d0 * (a * c))
    else
        tmp = sqrt((f * (a - b_m))) * (sqrt(2.0d0) / -b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.4e-55) {
		tmp = Math.sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C));
	} else {
		tmp = Math.sqrt((F * (A - B_m))) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 4.4e-55:
		tmp = math.sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C))
	else:
		tmp = math.sqrt((F * (A - B_m))) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4.4e-55)
		tmp = Float64(sqrt(Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * Float64(F * Float64(A + A))))))) / Float64(4.0 * Float64(A * C)));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - B_m))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 4.4e-55)
		tmp = sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C));
	else
		tmp = sqrt((F * (A - B_m))) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 4.4e-55], N[(N[Sqrt[N[(2.0 * N[(-4.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.3999999999999999e-55

   1. Initial program 21.1%

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

   3. Simplified 23.8%

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

   5. Taylor expanded in C around inf 13.3%

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

      1. mul-1-neg 13.3%

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

   7. Simplified 13.3%

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

   8. Taylor expanded in A around inf 13.5%

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

   if 4.3999999999999999e-55 < B

   1. Initial program 17.1%

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

   3. Taylor expanded in C around 0 26.9%

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

      1. mul-1-neg 26.9%

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

      2. +-commutative 26.9%

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

      3. unpow2 26.9%

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

      4. unpow2 26.9%

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

      5. hypot-define 52.0%

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

   5. Simplified 52.0%

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

   6. Taylor expanded in A around 0 47.0%

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

      1. neg-mul-1 47.0%

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

      2. unsub-neg 47.0%

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

   8. Simplified 47.0%

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

4. Final simplification 23.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.4 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(-B\_m\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5e-55)
   (/ (sqrt (* 2.0 (* -4.0 (* A (* C (* F (+ A A))))))) (* 4.0 (* A C)))
   (* (sqrt (* F (- B_m))) (/ (sqrt 2.0) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5e-55) {
		tmp = sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C));
	} else {
		tmp = sqrt((F * -B_m)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 5d-55) then
        tmp = sqrt((2.0d0 * ((-4.0d0) * (a * (c * (f * (a + a))))))) / (4.0d0 * (a * c))
    else
        tmp = sqrt((f * -b_m)) * (sqrt(2.0d0) / -b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5e-55) {
		tmp = Math.sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C));
	} else {
		tmp = Math.sqrt((F * -B_m)) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 5e-55:
		tmp = math.sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C))
	else:
		tmp = math.sqrt((F * -B_m)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5e-55)
		tmp = Float64(sqrt(Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * Float64(F * Float64(A + A))))))) / Float64(4.0 * Float64(A * C)));
	else
		tmp = Float64(sqrt(Float64(F * Float64(-B_m))) * Float64(sqrt(2.0) / Float64(-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 <= 5e-55)
		tmp = sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C));
	else
		tmp = sqrt((F * -B_m)) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 5e-55], N[(N[Sqrt[N[(2.0 * N[(-4.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * (-B$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.0000000000000002e-55

   1. Initial program 21.1%

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

   3. Simplified 23.8%

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

   5. Taylor expanded in C around inf 13.3%

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

      1. mul-1-neg 13.3%

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

   7. Simplified 13.3%

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

   8. Taylor expanded in A around inf 13.5%

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

   if 5.0000000000000002e-55 < B

   1. Initial program 17.1%

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

   3. Taylor expanded in C around 0 26.9%

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

      1. mul-1-neg 26.9%

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

      2. +-commutative 26.9%

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

      3. unpow2 26.9%

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

      4. unpow2 26.9%

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

      5. hypot-define 52.0%

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

   5. Simplified 52.0%

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

   6. Taylor expanded in A around 0 47.1%

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

      1. associate-*r* 47.1%

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

      2. neg-mul-1 47.1%

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

   8. Simplified 47.1%

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

4. Final simplification 23.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(-B\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 24.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)} \end{array} \]

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 (* -4.0 (* A (* C (* F (+ A A))))))) (* 4.0 (* A C))))

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 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C));
}

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 * ((-4.0d0) * (a * (c * (f * (a + a))))))) / (4.0d0 * (a * c))
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 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C));
}

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 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C))

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(-4.0 * Float64(A * Float64(C * Float64(F * Float64(A + A))))))) / Float64(4.0 * Float64(A * C)))
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 * (-4.0 * (A * (C * (F * (A + A))))))) / (4.0 * (A * C));
end

Wolfram

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

Derivation

1. Initial program 20.0%

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

3. Simplified 21.6%

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

5. Taylor expanded in C around inf 10.6%

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

   1. mul-1-neg 10.6%

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

7. Simplified 10.6%

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

8. Taylor expanded in A around inf 10.7%

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

9. Final simplification 10.7%

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

Alternative 9: 1.0% accurate, 5.9× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (* -2.0 (/ (sqrt (* C 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 -2.0 * (sqrt((C * 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 = (-2.0d0) * (sqrt((c * 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 -2.0 * (Math.sqrt((C * 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 -2.0 * (math.sqrt((C * 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(-2.0 * Float64(sqrt(Float64(C * 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 = -2.0 * (sqrt((C * 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[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in C around -inf 16.7%

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

4. Taylor expanded in B around inf 4.3%

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

   1. associate-*l/ 4.3%

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

   2. *-lft-identity 4.3%

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

   3. *-commutative 4.3%

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

6. Simplified 4.3%

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

7. Final simplification 4.3%

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

Reproduce

herbie shell --seed 2024053 
(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))))