ABCF->ab-angle b

Percentage Accurate: 19.5% → 43.1%

Time: 27.9s

Alternatives: 10

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.5% 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: 43.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{hypot}\left(B_m, \sqrt{t_0}\right)\\ t_2 := \mathsf{fma}\left(B_m, B_m, t_0\right)\\ \mathbf{if}\;C \leq -1.45 \cdot 10^{-305}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \mathbf{elif}\;C \leq 650:\\ \;\;\;\;\frac{t_1}{\frac{t_1}{\sqrt{F \cdot \left(2 \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B_m, A - C\right)\right)\right)}}} \cdot \frac{-1}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_2\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{-0.5 \cdot \left(\left({A}^{2} + {B_m}^{2}\right) - {\left(-A\right)}^{2}\right)}{C}\right)\right)\right)}}{t_2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* A (* C -4.0)))
        (t_1 (hypot B_m (sqrt t_0)))
        (t_2 (fma B_m B_m t_0)))
   (if (<= C -1.45e-305)
     (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (- A (hypot B_m A)))))
     (if (<= C 650.0)
       (*
        (/ t_1 (/ t_1 (sqrt (* F (* 2.0 (- (+ C A) (hypot B_m (- A C))))))))
        (/ -1.0 t_1))
       (/
        (-
         (sqrt
          (*
           (* F t_2)
           (*
            2.0
            (+
             A
             (+
              A
              (/
               (* -0.5 (- (+ (pow A 2.0) (pow B_m 2.0)) (pow (- A) 2.0)))
               C)))))))
        t_2)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = A * (C * -4.0);
	double t_1 = hypot(B_m, sqrt(t_0));
	double t_2 = fma(B_m, B_m, t_0);
	double tmp;
	if (C <= -1.45e-305) {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (A - hypot(B_m, A))));
	} else if (C <= 650.0) {
		tmp = (t_1 / (t_1 / sqrt((F * (2.0 * ((C + A) - hypot(B_m, (A - C)))))))) * (-1.0 / t_1);
	} else {
		tmp = -sqrt(((F * t_2) * (2.0 * (A + (A + ((-0.5 * ((pow(A, 2.0) + pow(B_m, 2.0)) - pow(-A, 2.0))) / C)))))) / t_2;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(A * Float64(C * -4.0))
	t_1 = hypot(B_m, sqrt(t_0))
	t_2 = fma(B_m, B_m, t_0)
	tmp = 0.0
	if (C <= -1.45e-305)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(A - hypot(B_m, A)))));
	elseif (C <= 650.0)
		tmp = Float64(Float64(t_1 / Float64(t_1 / sqrt(Float64(F * Float64(2.0 * Float64(Float64(C + A) - hypot(B_m, Float64(A - C)))))))) * Float64(-1.0 / t_1));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_2) * Float64(2.0 * Float64(A + Float64(A + Float64(Float64(-0.5 * Float64(Float64((A ^ 2.0) + (B_m ^ 2.0)) - (Float64(-A) ^ 2.0))) / C))))))) / t_2);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B$95$m ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(B$95$m * B$95$m + t$95$0), $MachinePrecision]}, If[LessEqual[C, -1.45e-305], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 650.0], N[(N[(t$95$1 / N[(t$95$1 / N[Sqrt[N[(F * N[(2.0 * N[(N[(C + A), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(F * t$95$2), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(N[(-0.5 * N[(N[(N[Power[A, 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - N[Power[(-A), 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.44999999999999994e-305

   1. Initial program 14.8%

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

   3. Simplified 17.9%

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

   4. Taylor expanded in C around 0 8.6%

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

      1. associate-*r* 8.6%

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

      2. mul-1-neg 8.6%

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

      3. +-commutative 8.6%

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

      4. unpow2 8.6%

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

      5. unpow2 8.6%

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

      6. hypot-def 15.1%

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

   6. Simplified 15.1%

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

   if -1.44999999999999994e-305 < C < 650

   1. Initial program 20.4%

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

   3. Simplified 29.0%

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

      1. neg-mul-1 29.0%

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

      2. add-sqr-sqrt 28.7%

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

      3. times-frac 28.7%

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

      4. fma-udef 28.7%

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

      5. add-sqr-sqrt 26.8%

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

      6. hypot-def 26.8%

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

   5. Applied egg-rr 43.3%

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

      1. *-commutative 43.3%

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

      2. associate-/l* 47.7%

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

   7. Simplified 47.7%

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

   if 650 < C

   1. Initial program 3.0%

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

   3. Simplified 8.5%

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

   4. Taylor expanded in C around inf 43.1%

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

      1. associate-*r/ 43.1%

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

      2. mul-1-neg 43.1%

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

   6. Simplified 43.1%

      \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \color{blue}{\left(A + \frac{-0.5 \cdot \left(\left({A}^{2} + {B}^{2}\right) - {\left(-A\right)}^{2}\right)}{C}\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.45 \cdot 10^{-305}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{elif}\;C \leq 650:\\ \;\;\;\;\frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\sqrt{F \cdot \left(2 \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}} \cdot \frac{-1}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{-0.5 \cdot \left(\left({A}^{2} + {B}^{2}\right) - {\left(-A\right)}^{2}\right)}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 2: 43.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{hypot}\left(B_m, \sqrt{t_0}\right)\\ t_2 := \mathsf{fma}\left(B_m, B_m, t_0\right)\\ \mathbf{if}\;C \leq 3.8 \cdot 10^{-305}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \mathbf{elif}\;C \leq 3000:\\ \;\;\;\;\frac{\frac{-t_1}{\frac{t_1}{\sqrt{F \cdot \left(2 \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B_m, A - C\right)\right)\right)}}}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_2\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{-0.5 \cdot \left(\left({A}^{2} + {B_m}^{2}\right) - {\left(-A\right)}^{2}\right)}{C}\right)\right)\right)}}{t_2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* A (* C -4.0)))
        (t_1 (hypot B_m (sqrt t_0)))
        (t_2 (fma B_m B_m t_0)))
   (if (<= C 3.8e-305)
     (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (- A (hypot B_m A)))))
     (if (<= C 3000.0)
       (/
        (/
         (- t_1)
         (/ t_1 (sqrt (* F (* 2.0 (- (+ C A) (hypot B_m (- A C))))))))
        t_1)
       (/
        (-
         (sqrt
          (*
           (* F t_2)
           (*
            2.0
            (+
             A
             (+
              A
              (/
               (* -0.5 (- (+ (pow A 2.0) (pow B_m 2.0)) (pow (- A) 2.0)))
               C)))))))
        t_2)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = A * (C * -4.0);
	double t_1 = hypot(B_m, sqrt(t_0));
	double t_2 = fma(B_m, B_m, t_0);
	double tmp;
	if (C <= 3.8e-305) {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (A - hypot(B_m, A))));
	} else if (C <= 3000.0) {
		tmp = (-t_1 / (t_1 / sqrt((F * (2.0 * ((C + A) - hypot(B_m, (A - C)))))))) / t_1;
	} else {
		tmp = -sqrt(((F * t_2) * (2.0 * (A + (A + ((-0.5 * ((pow(A, 2.0) + pow(B_m, 2.0)) - pow(-A, 2.0))) / C)))))) / t_2;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(A * Float64(C * -4.0))
	t_1 = hypot(B_m, sqrt(t_0))
	t_2 = fma(B_m, B_m, t_0)
	tmp = 0.0
	if (C <= 3.8e-305)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(A - hypot(B_m, A)))));
	elseif (C <= 3000.0)
		tmp = Float64(Float64(Float64(-t_1) / Float64(t_1 / sqrt(Float64(F * Float64(2.0 * Float64(Float64(C + A) - hypot(B_m, Float64(A - C)))))))) / t_1);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_2) * Float64(2.0 * Float64(A + Float64(A + Float64(Float64(-0.5 * Float64(Float64((A ^ 2.0) + (B_m ^ 2.0)) - (Float64(-A) ^ 2.0))) / C))))))) / t_2);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B$95$m ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(B$95$m * B$95$m + t$95$0), $MachinePrecision]}, If[LessEqual[C, 3.8e-305], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 3000.0], N[(N[((-t$95$1) / N[(t$95$1 / N[Sqrt[N[(F * N[(2.0 * N[(N[(C + A), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[((-N[Sqrt[N[(N[(F * t$95$2), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(N[(-0.5 * N[(N[(N[Power[A, 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - N[Power[(-A), 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if C < 3.8e-305

   1. Initial program 14.6%

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

   3. Simplified 17.7%

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

   4. Taylor expanded in C around 0 9.3%

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

      1. associate-*r* 9.3%

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

      2. mul-1-neg 9.3%

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

      3. +-commutative 9.3%

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

      4. unpow2 9.3%

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

      5. unpow2 9.3%

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

      6. hypot-def 15.6%

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

   6. Simplified 15.6%

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

   if 3.8e-305 < C < 3e3

   1. Initial program 20.9%

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

   3. Simplified 29.8%

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

      1. neg-mul-1 29.8%

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

      2. add-sqr-sqrt 29.5%

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

      3. times-frac 29.5%

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

      4. fma-udef 29.6%

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

      5. add-sqr-sqrt 27.6%

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

      6. hypot-def 27.6%

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

   5. Applied egg-rr 43.1%

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

      1. associate-*l/ 43.1%

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

      2. mul-1-neg 43.1%

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

      3. associate-/l* 47.6%

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

   7. Simplified 47.6%

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

   if 3e3 < C

   1. Initial program 3.0%

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

   3. Simplified 8.5%

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

   4. Taylor expanded in C around inf 43.1%

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

      1. associate-*r/ 43.1%

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

      2. mul-1-neg 43.1%

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

   6. Simplified 43.1%

      \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \color{blue}{\left(A + \frac{-0.5 \cdot \left(\left({A}^{2} + {B}^{2}\right) - {\left(-A\right)}^{2}\right)}{C}\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 3.8 \cdot 10^{-305}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{elif}\;C \leq 3000:\\ \;\;\;\;\frac{\frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\sqrt{F \cdot \left(2 \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{-0.5 \cdot \left(\left({A}^{2} + {B}^{2}\right) - {\left(-A\right)}^{2}\right)}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 3: 46.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B_m \leq 25000000000000:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_0\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{t_0}\\ \mathbf{elif}\;B_m \leq 1.55 \cdot 10^{+88}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \end{array} \end{array} \]

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 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 25000000000000.0)
     (/ (- (sqrt (* (* F t_0) (* 2.0 (+ A A))))) t_0)
     (if (<= B_m 1.55e+88)
       (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C))))))
       (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (- A (hypot B_m A)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 25000000000000.0) {
		tmp = -sqrt(((F * t_0) * (2.0 * (A + A)))) / t_0;
	} else if (B_m <= 1.55e+88) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	} else {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 25000000000000.0)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + A))))) / t_0);
	elseif (B_m <= 1.55e+88)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C))))));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(A - hypot(B_m, A)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 25000000000000.0], N[((-N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 1.55e+88], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.5e13

   1. Initial program 13.4%

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

   3. Simplified 22.4%

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

   4. Taylor expanded in C around inf 18.6%

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

   if 2.5e13 < B < 1.5500000000000001e88

   1. Initial program 31.8%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in A around 0 37.1%

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

      1. associate-*r* 37.1%

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

      2. mul-1-neg 37.1%

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

      3. unpow2 37.1%

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

      4. unpow2 37.1%

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

      5. hypot-def 42.8%

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

   6. Simplified 42.8%

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

   7. Taylor expanded in C around inf 26.0%

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

   if 1.5500000000000001e88 < B

   1. Initial program 3.2%

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

   3. Simplified 0.7%

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

   4. Taylor expanded in C around 0 10.5%

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

      1. associate-*r* 10.5%

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

      2. mul-1-neg 10.5%

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

      3. +-commutative 10.5%

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

      4. unpow2 10.5%

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

      5. unpow2 10.5%

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

      6. hypot-def 51.9%

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

   6. Simplified 51.9%

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

4. Final simplification 23.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 25000000000000:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+88}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 4: 46.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B_m \leq 54000000000000:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(F \cdot \left(A \cdot 4\right)\right)}}{t_0}\\ \mathbf{elif}\;B_m \leq 2.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \end{array} \end{array} \]

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 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 54000000000000.0)
     (/ (- (sqrt (* t_0 (* F (* A 4.0))))) t_0)
     (if (<= B_m 2.4e+89)
       (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C))))))
       (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (- A (hypot B_m A)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 54000000000000.0) {
		tmp = -sqrt((t_0 * (F * (A * 4.0)))) / t_0;
	} else if (B_m <= 2.4e+89) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	} else {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 54000000000000.0)
		tmp = Float64(Float64(-sqrt(Float64(t_0 * Float64(F * Float64(A * 4.0))))) / t_0);
	elseif (B_m <= 2.4e+89)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C))))));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(A - hypot(B_m, A)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 54000000000000.0], N[((-N[Sqrt[N[(t$95$0 * N[(F * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 2.4e+89], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 5.4e13

   1. Initial program 13.4%

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

   3. Simplified 22.4%

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

   4. Taylor expanded in C around inf 18.6%

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

      1. expm1-log1p-u 17.8%

         \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + A\right)\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. expm1-udef 10.2%

         \[\leadsto \frac{-\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + A\right)\right)}\right)} - 1\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. associate-*l* 9.9%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(A + A\right)\right)\right)}}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. count-2 9.9%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \color{blue}{\left(2 \cdot A\right)}\right)\right)}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Applied egg-rr 9.9%

      \[\leadsto \frac{-\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(2 \cdot A\right)\right)\right)}\right)} - 1\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-def 16.6%

         \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(2 \cdot A\right)\right)\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. expm1-log1p 17.4%

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

      3. associate-*r* 17.4%

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

      4. metadata-eval 17.4%

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

   8. Simplified 17.4%

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

   if 5.4e13 < B < 2.40000000000000004e89

   1. Initial program 31.8%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in A around 0 37.1%

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

      1. associate-*r* 37.1%

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

      2. mul-1-neg 37.1%

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

      3. unpow2 37.1%

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

      4. unpow2 37.1%

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

      5. hypot-def 42.8%

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

   6. Simplified 42.8%

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

   7. Taylor expanded in C around inf 26.0%

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

   if 2.40000000000000004e89 < B

   1. Initial program 3.2%

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

   3. Simplified 0.7%

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

   4. Taylor expanded in C around 0 10.5%

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

      1. associate-*r* 10.5%

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

      2. mul-1-neg 10.5%

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

      3. +-commutative 10.5%

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

      4. unpow2 10.5%

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

      5. unpow2 10.5%

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

      6. hypot-def 51.9%

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

   6. Simplified 51.9%

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

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 54000000000000:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(A \cdot 4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 5: 43.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B_m \leq 8 \cdot 10^{-68}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)}\\ \mathbf{elif}\;B_m \leq 4.6 \cdot 10^{+88}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \end{array} \end{array} \]

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
 (if (<= B_m 8e-68)
   (/
    (- (sqrt (* -8.0 (* A (* C (* F (+ A A)))))))
    (fma A (* C -4.0) (pow B_m 2.0)))
   (if (<= B_m 4.6e+88)
     (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C))))))
     (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (- A (hypot B_m A))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 8e-68) {
		tmp = -sqrt((-8.0 * (A * (C * (F * (A + A)))))) / fma(A, (C * -4.0), pow(B_m, 2.0));
	} else if (B_m <= 4.6e+88) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	} else {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (A - hypot(B_m, A))));
	}
	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 <= 8e-68)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A))))))) / fma(A, Float64(C * -4.0), (B_m ^ 2.0)));
	elseif (B_m <= 4.6e+88)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C))))));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(A - hypot(B_m, A)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 8e-68], N[((-N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.6e+88], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 8.00000000000000053e-68

   1. Initial program 12.5%

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

   3. Simplified 14.3%

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

   4. Taylor expanded in C around inf 17.2%

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

   if 8.00000000000000053e-68 < B < 4.6000000000000003e88

   1. Initial program 31.2%

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

   3. Simplified 38.0%

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

   4. Taylor expanded in A around 0 34.9%

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

      1. associate-*r* 34.9%

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

      2. mul-1-neg 34.9%

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

      3. unpow2 34.9%

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

      4. unpow2 34.9%

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

      5. hypot-def 41.7%

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

   6. Simplified 41.7%

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

   7. Taylor expanded in C around inf 21.0%

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

   if 4.6000000000000003e88 < B

   1. Initial program 3.2%

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

   3. Simplified 0.7%

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

   4. Taylor expanded in C around 0 10.5%

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

      1. associate-*r* 10.5%

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

      2. mul-1-neg 10.5%

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

      3. +-commutative 10.5%

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

      4. unpow2 10.5%

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

      5. unpow2 10.5%

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

      6. hypot-def 51.9%

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

   6. Simplified 51.9%

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

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8 \cdot 10^{-68}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{+88}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 6: 41.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B_m \leq 2.05 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(A + A\right)\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B_m \leq 1.12 \cdot 10^{+89}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \end{array} \end{array} \]

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
 (if (<= B_m 2.05e-70)
   (/
    (- (sqrt (* (* 2.0 (+ A A)) (* (* A -4.0) (* C F)))))
    (fma B_m B_m (* A (* C -4.0))))
   (if (<= B_m 1.12e+89)
     (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C))))))
     (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (- A (hypot B_m A))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.05e-70) {
		tmp = -sqrt(((2.0 * (A + A)) * ((A * -4.0) * (C * F)))) / fma(B_m, B_m, (A * (C * -4.0)));
	} else if (B_m <= 1.12e+89) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	} else {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (A - hypot(B_m, A))));
	}
	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 <= 2.05e-70)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(A + A)) * Float64(Float64(A * -4.0) * Float64(C * F))))) / fma(B_m, B_m, Float64(A * Float64(C * -4.0))));
	elseif (B_m <= 1.12e+89)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C))))));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(A - hypot(B_m, A)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2.05e-70], N[((-N[Sqrt[N[(N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision] * N[(N[(A * -4.0), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.12e+89], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.04999999999999989e-70

   1. Initial program 12.5%

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

   3. Simplified 20.5%

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

   4. Taylor expanded in C around inf 18.0%

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

   5. Taylor expanded in B around 0 15.8%

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

      1. associate-*r* 15.8%

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

      2. *-commutative 15.8%

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

   7. Simplified 15.8%

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

   if 2.04999999999999989e-70 < B < 1.11999999999999995e89

   1. Initial program 31.2%

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

   3. Simplified 38.0%

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

   4. Taylor expanded in A around 0 34.9%

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

      1. associate-*r* 34.9%

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

      2. mul-1-neg 34.9%

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

      3. unpow2 34.9%

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

      4. unpow2 34.9%

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

      5. hypot-def 41.7%

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

   6. Simplified 41.7%

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

   7. Taylor expanded in C around inf 21.0%

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

   if 1.11999999999999995e89 < B

   1. Initial program 3.2%

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

   3. Simplified 0.7%

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

   4. Taylor expanded in C around 0 10.5%

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

      1. associate-*r* 10.5%

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

      2. mul-1-neg 10.5%

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

      3. +-commutative 10.5%

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

      4. unpow2 10.5%

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

      5. unpow2 10.5%

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

      6. hypot-def 51.9%

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

   6. Simplified 51.9%

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

4. Final simplification 21.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.05 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(A + A\right)\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.12 \cdot 10^{+89}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 7: 42.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B_m \leq 1.85 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(A + A\right)\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \end{array} \end{array} \]

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
 (if (<= B_m 1.85e-90)
   (/
    (- (sqrt (* (* 2.0 (+ A A)) (* (* A -4.0) (* C F)))))
    (fma B_m B_m (* A (* C -4.0))))
   (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (- A (hypot B_m A)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.85e-90) {
		tmp = -sqrt(((2.0 * (A + A)) * ((A * -4.0) * (C * F)))) / fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.85e-90)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(A + A)) * Float64(Float64(A * -4.0) * Float64(C * F))))) / fma(B_m, B_m, Float64(A * Float64(C * -4.0))));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(A - hypot(B_m, A)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.85e-90], N[((-N[Sqrt[N[(N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision] * N[(N[(A * -4.0), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.85000000000000009e-90

   1. Initial program 12.2%

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

   3. Simplified 19.8%

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

   4. Taylor expanded in C around inf 17.7%

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

   5. Taylor expanded in B around 0 15.5%

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

      1. associate-*r* 15.5%

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

      2. *-commutative 15.5%

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

   7. Simplified 15.5%

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

   if 1.85000000000000009e-90 < B

   1. Initial program 16.8%

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

   3. Simplified 18.5%

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

   4. Taylor expanded in C around 0 20.1%

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

      1. associate-*r* 20.1%

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

      2. mul-1-neg 20.1%

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

      3. +-commutative 20.1%

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

      4. unpow2 20.1%

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

      5. unpow2 20.1%

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

      6. hypot-def 41.3%

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

   6. Simplified 41.3%

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

4. Final simplification 22.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.85 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(A + A\right)\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 8: 38.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B_m \leq 25000000000000:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(A + A\right)\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{B_m \cdot \left(-F\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 25000000000000.0)
   (/
    (- (sqrt (* (* 2.0 (+ A A)) (* (* A -4.0) (* C F)))))
    (fma B_m B_m (* A (* C -4.0))))
   (* (/ (- (sqrt 2.0)) B_m) (sqrt (* B_m (- F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 25000000000000.0) {
		tmp = -sqrt(((2.0 * (A + A)) * ((A * -4.0) * (C * F)))) / fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = (-sqrt(2.0) / B_m) * sqrt((B_m * -F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 25000000000000.0)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(A + A)) * Float64(Float64(A * -4.0) * Float64(C * F))))) / fma(B_m, B_m, Float64(A * Float64(C * -4.0))));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(B_m * Float64(-F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 25000000000000.0], N[((-N[Sqrt[N[(N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision] * N[(N[(A * -4.0), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(B$95$m * (-F)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.5e13

   1. Initial program 13.4%

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

   3. Simplified 22.4%

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

   4. Taylor expanded in C around inf 18.6%

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

   5. Taylor expanded in B around 0 15.1%

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

      1. associate-*r* 15.1%

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

      2. *-commutative 15.1%

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

   7. Simplified 15.1%

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

   if 2.5e13 < B

   1. Initial program 13.4%

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

   3. Simplified 12.0%

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

   4. Taylor expanded in A around 0 20.3%

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

      1. associate-*r* 20.3%

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

      2. mul-1-neg 20.3%

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

      3. unpow2 20.3%

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

      4. unpow2 20.3%

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

      5. hypot-def 42.8%

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

   6. Simplified 42.8%

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

   7. Taylor expanded in C around 0 37.7%

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

      1. mul-1-neg 37.7%

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

   9. Simplified 37.7%

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

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 25000000000000:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(A + A\right)\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{B \cdot \left(-F\right)}\\ \end{array} \]

Alternative 9: 26.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -4.5 \cdot 10^{+75}:\\ \;\;\;\;-2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{B_m \cdot \left(-F\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -4.5e+75)
   (* -2.0 (* (sqrt (* F A)) (/ 1.0 B_m)))
   (* (/ (- (sqrt 2.0)) B_m) (sqrt (* B_m (- F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -4.5e+75) {
		tmp = -2.0 * (sqrt((F * A)) * (1.0 / B_m));
	} else {
		tmp = (-sqrt(2.0) / B_m) * sqrt((B_m * -F));
	}
	return tmp;
}

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (a <= (-4.5d+75)) then
        tmp = (-2.0d0) * (sqrt((f * a)) * (1.0d0 / b_m))
    else
        tmp = (-sqrt(2.0d0) / b_m) * sqrt((b_m * -f))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -4.5e+75) {
		tmp = -2.0 * (Math.sqrt((F * A)) * (1.0 / B_m));
	} else {
		tmp = (-Math.sqrt(2.0) / B_m) * Math.sqrt((B_m * -F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if A <= -4.5e+75:
		tmp = -2.0 * (math.sqrt((F * A)) * (1.0 / B_m))
	else:
		tmp = (-math.sqrt(2.0) / B_m) * math.sqrt((B_m * -F))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -4.5e+75)
		tmp = Float64(-2.0 * Float64(sqrt(Float64(F * A)) * Float64(1.0 / B_m)));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(B_m * Float64(-F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (A <= -4.5e+75)
		tmp = -2.0 * (sqrt((F * A)) * (1.0 / B_m));
	else
		tmp = (-sqrt(2.0) / B_m) * sqrt((B_m * -F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -4.5e+75], N[(-2.0 * N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(B$95$m * (-F)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.5000000000000004e75

   1. Initial program 13.1%

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

   3. Simplified 30.9%

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

   4. Taylor expanded in C around inf 32.7%

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

   5. Taylor expanded in B around inf 9.1%

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

   if -4.5000000000000004e75 < A

   1. Initial program 13.5%

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

   3. Simplified 15.2%

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

   4. Taylor expanded in A around 0 9.9%

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

      1. associate-*r* 9.9%

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

      2. mul-1-neg 9.9%

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

      3. unpow2 9.9%

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

      4. unpow2 9.9%

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

      5. hypot-def 16.3%

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

   6. Simplified 16.3%

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

   7. Taylor expanded in C around 0 12.8%

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

      1. mul-1-neg 12.8%

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

   9. Simplified 12.8%

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

4. Final simplification 12.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4.5 \cdot 10^{+75}:\\ \;\;\;\;-2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{B \cdot \left(-F\right)}\\ \end{array} \]

Alternative 10: 9.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B_m}\right) \end{array} \]

FPCore

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

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -2.0 * (sqrt((F * A)) * (1.0 / B_m));
}

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-2.0d0) * (sqrt((f * a)) * (1.0d0 / b_m))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -2.0 * (Math.sqrt((F * A)) * (1.0 / B_m));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -2.0 * (math.sqrt((F * A)) * (1.0 / B_m))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-2.0 * Float64(sqrt(Float64(F * A)) * Float64(1.0 / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -2.0 * (sqrt((F * A)) * (1.0 / B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(-2.0 * N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.4%

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

3. Simplified 21.3%

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

4. Taylor expanded in C around inf 15.2%

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

5. Taylor expanded in B around inf 3.3%

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

6. Final simplification 3.3%

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

Reproduce

herbie shell --seed 2023332 
(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))))