ABCF->ab-angle b

Percentage Accurate: 18.3% → 47.8%

Time: 27.0s

Alternatives: 14

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: 18.3% 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: 47.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\\ t_1 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_1}\\ t_3 := 2 \cdot t_0\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B_m, A - C\right)\right)\right)} \cdot \left(-\sqrt{t_3}\right)}{t_0}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(t_3 \cdot \left(A + \left(A + -0.5 \cdot \frac{{A}^{2} + \left({B_m}^{2} - {\left(-A\right)}^{2}\right)}{C}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{-{\left(e^{0.25 \cdot \left(\log \left(-16 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)}\right)}^{2}}{\mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B_m}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \end{array} \end{array} \]

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 A (* C -4.0) (pow B_m 2.0)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0)))))))
          t_1))
        (t_3 (* 2.0 t_0)))
   (if (<= t_2 -1e-150)
     (/ (* (sqrt (* F (+ A (- C (hypot B_m (- A C)))))) (- (sqrt t_3))) t_0)
     (if (<= t_2 2e+194)
       (/
        (-
         (sqrt
          (*
           F
           (*
            t_3
            (+
             A
             (+
              A
              (*
               -0.5
               (/ (+ (pow A 2.0) (- (pow B_m 2.0) (pow (- A) 2.0))) C))))))))
        t_0)
       (if (<= t_2 INFINITY)
         (/
          (-
           (pow
            (exp
             (* 0.25 (+ (log (* -16.0 (* C F))) (* -2.0 (log (/ -1.0 A))))))
            2.0))
          (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 t_0 = fma(A, (C * -4.0), pow(B_m, 2.0));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_1;
	double t_3 = 2.0 * t_0;
	double tmp;
	if (t_2 <= -1e-150) {
		tmp = (sqrt((F * (A + (C - hypot(B_m, (A - C)))))) * -sqrt(t_3)) / t_0;
	} else if (t_2 <= 2e+194) {
		tmp = -sqrt((F * (t_3 * (A + (A + (-0.5 * ((pow(A, 2.0) + (pow(B_m, 2.0) - pow(-A, 2.0))) / C))))))) / t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = -pow(exp((0.25 * (log((-16.0 * (C * F))) + (-2.0 * log((-1.0 / A)))))), 2.0) / 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)
	t_0 = fma(A, Float64(C * -4.0), (B_m ^ 2.0))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_1)
	t_3 = Float64(2.0 * t_0)
	tmp = 0.0
	if (t_2 <= -1e-150)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A + Float64(C - hypot(B_m, Float64(A - C)))))) * Float64(-sqrt(t_3))) / t_0);
	elseif (t_2 <= 2e+194)
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(t_3 * Float64(A + Float64(A + Float64(-0.5 * Float64(Float64((A ^ 2.0) + Float64((B_m ^ 2.0) - (Float64(-A) ^ 2.0))) / C)))))))) / t_0);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(-(exp(Float64(0.25 * Float64(log(Float64(-16.0 * Float64(C * F))) + Float64(-2.0 * log(Float64(-1.0 / A)))))) ^ 2.0)) / 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_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-150], N[(N[(N[Sqrt[N[(F * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[t$95$3], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$2, 2e+194], N[((-N[Sqrt[N[(F * N[(t$95$3 * N[(A + N[(A + N[(-0.5 * N[(N[(N[Power[A, 2.0], $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[Power[(-A), 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[((-N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[N[(-1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $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 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -1.00000000000000001e-150

   1. Initial program 44.3%

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

   3. Simplified 42.3%

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

      1. pow1/2 42.3%

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

      2. associate-*r* 55.0%

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

      3. unpow-prod-down 70.7%

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

      4. pow1/2 70.7%

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

   5. Applied egg-rr 70.7%

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

      1. unpow1/2 70.7%

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

      2. associate-+r- 70.0%

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

      3. +-commutative 70.0%

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

      4. associate--l+ 70.9%

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

   7. Simplified 70.9%

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

   if -1.00000000000000001e-150 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 1.99999999999999989e194

   1. Initial program 32.4%

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

   3. Simplified 33.8%

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

   4. Taylor expanded in C around inf 25.9%

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

      1. associate--l+ 25.9%

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

      2. associate--l+ 25.9%

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

      3. mul-1-neg 25.9%

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

      4. mul-1-neg 25.9%

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

   6. Simplified 25.9%

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

   if 1.99999999999999989e194 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 3.5%

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

   3. Simplified 26.5%

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

      1. add-sqr-sqrt 26.4%

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

      2. pow2 26.4%

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

      3. pow1/2 26.4%

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

      4. sqrt-pow1 26.4%

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

      5. associate-*l* 26.4%

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

      6. associate-+r- 26.4%

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

      7. metadata-eval 26.4%

         \[\leadsto \frac{-{\left({\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 26.4%

      \[\leadsto \frac{-\color{blue}{{\left({\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in A around -inf 47.0%

      \[\leadsto \frac{-{\color{blue}{\left(e^{0.25 \cdot \left(\log \left(-16 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)}\right)}}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

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

   1. Initial program 0.0%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in C around 0 2.3%

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

      1. associate-*r* 2.3%

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

      2. mul-1-neg 2.3%

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

      3. +-commutative 2.3%

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

      4. unpow2 2.3%

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

      5. unpow2 2.3%

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

      6. hypot-def 16.3%

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

   6. Simplified 16.3%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\right)}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(\left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{{A}^{2} + \left({B}^{2} - {\left(-A\right)}^{2}\right)}{C}\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{-{\left(e^{0.25 \cdot \left(\log \left(-16 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 2: 45.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := 2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B_m, A - C\right)\right)\right)\\ t_2 := \mathsf{fma}\left(B_m, B_m, t_0\right)\\ t_3 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_3}\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\frac{-\mathsf{hypot}\left(B_m, \sqrt{t_0}\right) \cdot \sqrt{F \cdot t_1}}{t_2}\\ \mathbf{elif}\;t_4 \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_2\right) \cdot t_1}}{t_2}\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\frac{-{\left({\left(t_2 \cdot \left(F \cdot \left(2 \cdot \left(2 \cdot A\right)\right)\right)\right)}^{0.25}\right)}^{2}}{t_2}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\frac{-{\left(e^{0.25 \cdot \left(\log \left(-16 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)}\right)}^{2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B_m}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \end{array} \end{array} \]

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 (* A (* C -4.0)))
        (t_1 (* 2.0 (+ A (- C (hypot B_m (- A C))))))
        (t_2 (fma B_m B_m t_0))
        (t_3 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0)))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ (- (* (hypot B_m (sqrt t_0)) (sqrt (* F t_1)))) t_2)
     (if (<= t_4 -1e-150)
       (/ (- (sqrt (* (* F t_2) t_1))) t_2)
       (if (<= t_4 2e+194)
         (/ (- (pow (pow (* t_2 (* F (* 2.0 (* 2.0 A)))) 0.25) 2.0)) t_2)
         (if (<= t_4 INFINITY)
           (/
            (-
             (pow
              (exp
               (* 0.25 (+ (log (* -16.0 (* C F))) (* -2.0 (log (/ -1.0 A))))))
              2.0))
            t_2)
           (* (- (/ (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 = A * (C * -4.0);
	double t_1 = 2.0 * (A + (C - hypot(B_m, (A - C))));
	double t_2 = fma(B_m, B_m, t_0);
	double t_3 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = -(hypot(B_m, sqrt(t_0)) * sqrt((F * t_1))) / t_2;
	} else if (t_4 <= -1e-150) {
		tmp = -sqrt(((F * t_2) * t_1)) / t_2;
	} else if (t_4 <= 2e+194) {
		tmp = -pow(pow((t_2 * (F * (2.0 * (2.0 * A)))), 0.25), 2.0) / t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = -pow(exp((0.25 * (log((-16.0 * (C * F))) + (-2.0 * log((-1.0 / A)))))), 2.0) / t_2;
	} 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 = Float64(A * Float64(C * -4.0))
	t_1 = Float64(2.0 * Float64(A + Float64(C - hypot(B_m, Float64(A - C)))))
	t_2 = fma(B_m, B_m, t_0)
	t_3 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(-Float64(hypot(B_m, sqrt(t_0)) * sqrt(Float64(F * t_1)))) / t_2);
	elseif (t_4 <= -1e-150)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_2) * t_1))) / t_2);
	elseif (t_4 <= 2e+194)
		tmp = Float64(Float64(-((Float64(t_2 * Float64(F * Float64(2.0 * Float64(2.0 * A)))) ^ 0.25) ^ 2.0)) / t_2);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(-(exp(Float64(0.25 * Float64(log(Float64(-16.0 * Float64(C * F))) + Float64(-2.0 * log(Float64(-1.0 / A)))))) ^ 2.0)) / t_2);
	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[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B$95$m * B$95$m + t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[((-N[(N[Sqrt[B$95$m ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, -1e-150], N[((-N[Sqrt[N[(N[(F * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 2e+194], N[((-N[Power[N[Power[N[(t$95$2 * N[(F * N[(2.0 * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[((-N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[N[(-1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]) / t$95$2), $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 5 regimes

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

   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 24.5%

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

      1. add-sqr-sqrt 24.5%

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

      2. sqr-neg 24.5%

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

      3. sqrt-unprod 0.7%

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

      4. add-sqr-sqrt 1.8%

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

      5. neg-sub0 1.8%

         \[\leadsto \frac{-\color{blue}{\left(0 - \sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. sub-neg 1.8%

         \[\leadsto \frac{-\color{blue}{\left(0 + \left(-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. add-sqr-sqrt 0.7%

         \[\leadsto \frac{-\left(0 + \color{blue}{\sqrt{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}} \cdot \sqrt{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 44.2%

      \[\leadsto \frac{-\color{blue}{\left(0 + \mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. +-lft-identity 44.2%

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

      2. associate--l+ 45.8%

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

   7. Simplified 45.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -1.00000000000000001e-150

   1. Initial program 96.0%

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

   3. Simplified 95.9%

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

   if -1.00000000000000001e-150 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 1.99999999999999989e194

   1. Initial program 32.4%

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

   3. Simplified 33.7%

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

      1. add-sqr-sqrt 33.5%

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

      2. pow2 33.5%

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

      3. pow1/2 33.5%

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

      4. sqrt-pow1 33.5%

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

      5. associate-*l* 35.8%

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

      6. associate-+r- 34.5%

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

      7. metadata-eval 34.5%

         \[\leadsto \frac{-{\left({\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 34.5%

      \[\leadsto \frac{-\color{blue}{{\left({\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in A around -inf 24.0%

      \[\leadsto \frac{-{\left({\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \color{blue}{\left(2 \cdot A\right)}\right)\right)\right)}^{0.25}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 1.99999999999999989e194 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 3.5%

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

   3. Simplified 26.5%

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

      1. add-sqr-sqrt 26.4%

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

      2. pow2 26.4%

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

      3. pow1/2 26.4%

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

      4. sqrt-pow1 26.4%

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

      5. associate-*l* 26.4%

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

      6. associate-+r- 26.4%

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

      7. metadata-eval 26.4%

         \[\leadsto \frac{-{\left({\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 26.4%

      \[\leadsto \frac{-\color{blue}{{\left({\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in A around -inf 47.0%

      \[\leadsto \frac{-{\color{blue}{\left(e^{0.25 \cdot \left(\log \left(-16 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)}\right)}}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

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

   1. Initial program 0.0%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in C around 0 2.3%

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

      1. associate-*r* 2.3%

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

      2. mul-1-neg 2.3%

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

      3. +-commutative 2.3%

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

      4. unpow2 2.3%

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

      5. unpow2 2.3%

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

      6. hypot-def 16.3%

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

   6. Simplified 16.3%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -\infty:\\ \;\;\;\;\frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{F \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\frac{-{\left({\left(\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)}^{0.25}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{-{\left(e^{0.25 \cdot \left(\log \left(-16 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 3: 47.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := 2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B_m, A - C\right)\right)\right)\\ t_2 := \mathsf{fma}\left(B_m, B_m, t_0\right)\\ t_3 := F \cdot t_2\\ t_4 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := \frac{-\sqrt{\left(2 \cdot \left(t_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_4}\\ \mathbf{if}\;t_5 \leq -\infty:\\ \;\;\;\;\frac{-\mathsf{hypot}\left(B_m, \sqrt{t_0}\right) \cdot \sqrt{F \cdot t_1}}{t_2}\\ \mathbf{elif}\;t_5 \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot t_1}}{t_2}\\ \mathbf{elif}\;t_5 \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{A}^{2} + \left({B_m}^{2} - {\left(-A\right)}^{2}\right)}{C}\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;\frac{-{\left(e^{0.25 \cdot \left(\log \left(-16 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)}\right)}^{2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B_m}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \end{array} \end{array} \]

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 (* A (* C -4.0)))
        (t_1 (* 2.0 (+ A (- C (hypot B_m (- A C))))))
        (t_2 (fma B_m B_m t_0))
        (t_3 (* F t_2))
        (t_4 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_5
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_4 F))
             (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0)))))))
          t_4)))
   (if (<= t_5 (- INFINITY))
     (/ (- (* (hypot B_m (sqrt t_0)) (sqrt (* F t_1)))) t_2)
     (if (<= t_5 -1e-150)
       (/ (- (sqrt (* t_3 t_1))) t_2)
       (if (<= t_5 2e+194)
         (/
          (-
           (sqrt
            (*
             t_3
             (*
              2.0
              (+
               A
               (+
                A
                (*
                 -0.5
                 (/ (+ (pow A 2.0) (- (pow B_m 2.0) (pow (- A) 2.0))) C))))))))
          t_2)
         (if (<= t_5 INFINITY)
           (/
            (-
             (pow
              (exp
               (* 0.25 (+ (log (* -16.0 (* C F))) (* -2.0 (log (/ -1.0 A))))))
              2.0))
            t_2)
           (* (- (/ (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 = A * (C * -4.0);
	double t_1 = 2.0 * (A + (C - hypot(B_m, (A - C))));
	double t_2 = fma(B_m, B_m, t_0);
	double t_3 = F * t_2;
	double t_4 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_5 = -sqrt(((2.0 * (t_4 * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_4;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = -(hypot(B_m, sqrt(t_0)) * sqrt((F * t_1))) / t_2;
	} else if (t_5 <= -1e-150) {
		tmp = -sqrt((t_3 * t_1)) / t_2;
	} else if (t_5 <= 2e+194) {
		tmp = -sqrt((t_3 * (2.0 * (A + (A + (-0.5 * ((pow(A, 2.0) + (pow(B_m, 2.0) - pow(-A, 2.0))) / C))))))) / t_2;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = -pow(exp((0.25 * (log((-16.0 * (C * F))) + (-2.0 * log((-1.0 / A)))))), 2.0) / t_2;
	} 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 = Float64(A * Float64(C * -4.0))
	t_1 = Float64(2.0 * Float64(A + Float64(C - hypot(B_m, Float64(A - C)))))
	t_2 = fma(B_m, B_m, t_0)
	t_3 = Float64(F * t_2)
	t_4 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_5 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_4)
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(Float64(-Float64(hypot(B_m, sqrt(t_0)) * sqrt(Float64(F * t_1)))) / t_2);
	elseif (t_5 <= -1e-150)
		tmp = Float64(Float64(-sqrt(Float64(t_3 * t_1))) / t_2);
	elseif (t_5 <= 2e+194)
		tmp = Float64(Float64(-sqrt(Float64(t_3 * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64(Float64((A ^ 2.0) + Float64((B_m ^ 2.0) - (Float64(-A) ^ 2.0))) / C)))))))) / t_2);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(-(exp(Float64(0.25 * Float64(log(Float64(-16.0 * Float64(C * F))) + Float64(-2.0 * log(Float64(-1.0 / A)))))) ^ 2.0)) / t_2);
	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[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B$95$m * B$95$m + t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(F * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[((-N[(N[Sqrt[B$95$m ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[t$95$5, -1e-150], N[((-N[Sqrt[N[(t$95$3 * t$95$1), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[t$95$5, 2e+194], N[((-N[Sqrt[N[(t$95$3 * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[(N[Power[A, 2.0], $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[Power[(-A), 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[((-N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[N[(-1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]) / t$95$2), $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 5 regimes

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

   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 24.5%

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

      1. add-sqr-sqrt 24.5%

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

      2. sqr-neg 24.5%

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

      3. sqrt-unprod 0.7%

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

      4. add-sqr-sqrt 1.8%

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

      5. neg-sub0 1.8%

         \[\leadsto \frac{-\color{blue}{\left(0 - \sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. sub-neg 1.8%

         \[\leadsto \frac{-\color{blue}{\left(0 + \left(-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. add-sqr-sqrt 0.7%

         \[\leadsto \frac{-\left(0 + \color{blue}{\sqrt{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}} \cdot \sqrt{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 44.2%

      \[\leadsto \frac{-\color{blue}{\left(0 + \mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   6. Step-by-step derivation

      1. +-lft-identity 44.2%

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

      2. associate--l+ 45.8%

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

   7. Simplified 45.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -1.00000000000000001e-150

   1. Initial program 96.0%

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

   3. Simplified 95.9%

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

   if -1.00000000000000001e-150 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 1.99999999999999989e194

   1. Initial program 32.4%

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

   3. Simplified 33.7%

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

   4. Taylor expanded in C around inf 23.9%

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

      1. associate--l+ 23.9%

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

      2. mul-1-neg 23.9%

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

   6. Simplified 23.9%

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

   if 1.99999999999999989e194 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 3.5%

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

   3. Simplified 26.5%

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

      1. add-sqr-sqrt 26.4%

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

      2. pow2 26.4%

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

      3. pow1/2 26.4%

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

      4. sqrt-pow1 26.4%

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

      5. associate-*l* 26.4%

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

      6. associate-+r- 26.4%

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

      7. metadata-eval 26.4%

         \[\leadsto \frac{-{\left({\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 26.4%

      \[\leadsto \frac{-\color{blue}{{\left({\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in A around -inf 47.0%

      \[\leadsto \frac{-{\color{blue}{\left(e^{0.25 \cdot \left(\log \left(-16 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)}\right)}}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

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

   1. Initial program 0.0%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in C around 0 2.3%

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

      1. associate-*r* 2.3%

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

      2. mul-1-neg 2.3%

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

      3. +-commutative 2.3%

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

      4. unpow2 2.3%

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

      5. unpow2 2.3%

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

      6. hypot-def 16.3%

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

   6. Simplified 16.3%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -\infty:\\ \;\;\;\;\frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{F \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\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 + -0.5 \cdot \frac{{A}^{2} + \left({B}^{2} - {\left(-A\right)}^{2}\right)}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{-{\left(e^{0.25 \cdot \left(\log \left(-16 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 4: 49.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\\ t_2 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_2}\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B_m, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot t_1}\right)}{t_1}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_0\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{A}^{2} + \left({B_m}^{2} - {\left(-A\right)}^{2}\right)}{C}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\frac{-{\left(e^{0.25 \cdot \left(\log \left(-16 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)}\right)}^{2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B_m}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \end{array} \end{array} \]

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))))
        (t_1 (fma A (* C -4.0) (pow B_m 2.0)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0)))))))
          t_2)))
   (if (<= t_3 -1e-150)
     (/
      (* (sqrt (* F (+ A (- C (hypot B_m (- A C)))))) (- (sqrt (* 2.0 t_1))))
      t_1)
     (if (<= t_3 2e+194)
       (/
        (-
         (sqrt
          (*
           (* F t_0)
           (*
            2.0
            (+
             A
             (+
              A
              (*
               -0.5
               (/ (+ (pow A 2.0) (- (pow B_m 2.0) (pow (- A) 2.0))) C))))))))
        t_0)
       (if (<= t_3 INFINITY)
         (/
          (-
           (pow
            (exp
             (* 0.25 (+ (log (* -16.0 (* C F))) (* -2.0 (log (/ -1.0 A))))))
            2.0))
          t_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 t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = fma(A, (C * -4.0), pow(B_m, 2.0));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -1e-150) {
		tmp = (sqrt((F * (A + (C - hypot(B_m, (A - C)))))) * -sqrt((2.0 * t_1))) / t_1;
	} else if (t_3 <= 2e+194) {
		tmp = -sqrt(((F * t_0) * (2.0 * (A + (A + (-0.5 * ((pow(A, 2.0) + (pow(B_m, 2.0) - pow(-A, 2.0))) / C))))))) / t_0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = -pow(exp((0.25 * (log((-16.0 * (C * F))) + (-2.0 * log((-1.0 / A)))))), 2.0) / t_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)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = fma(A, Float64(C * -4.0), (B_m ^ 2.0))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_2)
	tmp = 0.0
	if (t_3 <= -1e-150)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A + Float64(C - hypot(B_m, Float64(A - C)))))) * Float64(-sqrt(Float64(2.0 * t_1)))) / t_1);
	elseif (t_3 <= 2e+194)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64(Float64((A ^ 2.0) + Float64((B_m ^ 2.0) - (Float64(-A) ^ 2.0))) / C)))))))) / t_0);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(-(exp(Float64(0.25 * Float64(log(Float64(-16.0 * Float64(C * F))) + Float64(-2.0 * log(Float64(-1.0 / A)))))) ^ 2.0)) / t_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_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-150], N[(N[(N[Sqrt[N[(F * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 2e+194], N[((-N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[(N[Power[A, 2.0], $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[Power[(-A), 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[((-N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[N[(-1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]) / t$95$0), $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 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -1.00000000000000001e-150

   1. Initial program 44.3%

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

   3. Simplified 42.3%

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

      1. pow1/2 42.3%

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

      2. associate-*r* 55.0%

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

      3. unpow-prod-down 70.7%

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

      4. pow1/2 70.7%

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

   5. Applied egg-rr 70.7%

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

      1. unpow1/2 70.7%

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

      2. associate-+r- 70.0%

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

      3. +-commutative 70.0%

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

      4. associate--l+ 70.9%

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

   7. Simplified 70.9%

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

   if -1.00000000000000001e-150 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 1.99999999999999989e194

   1. Initial program 32.4%

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

   3. Simplified 33.7%

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

   4. Taylor expanded in C around inf 23.9%

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

      1. associate--l+ 23.9%

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

      2. mul-1-neg 23.9%

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

   6. Simplified 23.9%

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

   if 1.99999999999999989e194 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 3.5%

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

   3. Simplified 26.5%

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

      1. add-sqr-sqrt 26.4%

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

      2. pow2 26.4%

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

      3. pow1/2 26.4%

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

      4. sqrt-pow1 26.4%

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

      5. associate-*l* 26.4%

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

      6. associate-+r- 26.4%

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

      7. metadata-eval 26.4%

         \[\leadsto \frac{-{\left({\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 26.4%

      \[\leadsto \frac{-\color{blue}{{\left({\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in A around -inf 47.0%

      \[\leadsto \frac{-{\color{blue}{\left(e^{0.25 \cdot \left(\log \left(-16 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)}\right)}}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

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

   1. Initial program 0.0%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in C around 0 2.3%

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

      1. associate-*r* 2.3%

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

      2. mul-1-neg 2.3%

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

      3. +-commutative 2.3%

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

      4. unpow2 2.3%

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

      5. unpow2 2.3%

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

      6. hypot-def 16.3%

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

   6. Simplified 16.3%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\right)}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{A}^{2} + \left({B}^{2} - {\left(-A\right)}^{2}\right)}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{-{\left(e^{0.25 \cdot \left(\log \left(-16 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 5: 44.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := A - \mathsf{hypot}\left(B_m, A\right)\\ \mathbf{if}\;{B_m}^{2} \leq 10^{-249}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)}\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{+122}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_0\right) \cdot \left(2 \cdot t_1\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B_m}\right) \cdot \sqrt{F \cdot t_1}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))) (t_1 (- A (hypot B_m A))))
   (if (<= (pow B_m 2.0) 1e-249)
     (/
      (- (sqrt (* -8.0 (* (* A C) (* F (+ A A))))))
      (fma A (* C -4.0) (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 1e+122)
       (/ (- (sqrt (* (* F t_0) (* 2.0 t_1)))) t_0)
       (* (- (/ (sqrt 2.0) B_m)) (sqrt (* F t_1)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = A - hypot(B_m, A);
	double tmp;
	if (pow(B_m, 2.0) <= 1e-249) {
		tmp = -sqrt((-8.0 * ((A * C) * (F * (A + A))))) / fma(A, (C * -4.0), pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 1e+122) {
		tmp = -sqrt(((F * t_0) * (2.0 * t_1))) / t_0;
	} else {
		tmp = -(sqrt(2.0) / B_m) * sqrt((F * t_1));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(A - hypot(B_m, A))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-249)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A)))))) / fma(A, Float64(C * -4.0), (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 1e+122)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_0) * Float64(2.0 * t_1)))) / t_0);
	else
		tmp = Float64(Float64(-Float64(sqrt(2.0) / B_m)) * sqrt(Float64(F * t_1)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-249], N[((-N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+122], N[((-N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]) * N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 1.00000000000000005e-249

   1. Initial program 18.7%

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

   3. Simplified 21.8%

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

   4. Taylor expanded in C around inf 20.3%

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

      1. associate-*r* 21.5%

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

      2. sub-neg 21.5%

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

      3. mul-1-neg 21.5%

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

      4. remove-double-neg 21.5%

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

   6. Simplified 21.5%

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

   if 1.00000000000000005e-249 < (pow.f64 B 2) < 1.00000000000000001e122

   1. Initial program 31.0%

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

   3. Simplified 38.8%

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

   4. Taylor expanded in C around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. +-commutative 22.7%

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

      3. unpow2 22.7%

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

      4. unpow2 22.7%

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

      5. hypot-def 27.2%

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

   6. Simplified 27.2%

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

   if 1.00000000000000001e122 < (pow.f64 B 2)

   1. Initial program 10.2%

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

   3. Simplified 9.2%

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

   4. Taylor expanded in C around 0 12.2%

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

      1. associate-*r* 12.2%

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

      2. mul-1-neg 12.2%

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

      3. +-commutative 12.2%

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

      4. unpow2 12.2%

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

      5. unpow2 12.2%

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

      6. hypot-def 28.6%

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

   6. Simplified 28.6%

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

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-249}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+122}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 6: 44.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B_m \leq 1.1 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)}\\ \mathbf{elif}\;B_m \leq 2.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_0\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B_m, A - C\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B_m}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \end{array} \end{array} \]

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 1.1e-124)
     (/
      (- (sqrt (* -8.0 (* (* A C) (* F (+ A A))))))
      (fma A (* C -4.0) (pow B_m 2.0)))
     (if (<= B_m 2.5e+61)
       (/ (- (sqrt (* (* F t_0) (* 2.0 (+ A (- C (hypot B_m (- A C)))))))) t_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 t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 1.1e-124) {
		tmp = -sqrt((-8.0 * ((A * C) * (F * (A + A))))) / fma(A, (C * -4.0), pow(B_m, 2.0));
	} else if (B_m <= 2.5e+61) {
		tmp = -sqrt(((F * t_0) * (2.0 * (A + (C - hypot(B_m, (A - C))))))) / t_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)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 1.1e-124)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A)))))) / fma(A, Float64(C * -4.0), (B_m ^ 2.0)));
	elseif (B_m <= 2.5e+61)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(C - hypot(B_m, Float64(A - C)))))))) / t_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_] := 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, 1.1e-124], N[((-N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $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, 2.5e+61], N[((-N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $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 < 1.0999999999999999e-124

   1. Initial program 17.8%

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

   3. Simplified 19.1%

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

   4. Taylor expanded in C around inf 12.3%

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

      1. associate-*r* 12.8%

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

      2. sub-neg 12.8%

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

      3. mul-1-neg 12.8%

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

      4. remove-double-neg 12.8%

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

   6. Simplified 12.8%

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

   if 1.0999999999999999e-124 < B < 2.50000000000000009e61

   1. Initial program 33.2%

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

   3. Simplified 43.4%

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

   if 2.50000000000000009e61 < B

   1. Initial program 13.9%

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

   3. Simplified 13.9%

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

   4. Taylor expanded in C around 0 24.2%

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

      1. associate-*r* 24.2%

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

      2. mul-1-neg 24.2%

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

      3. +-commutative 24.2%

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

      4. unpow2 24.2%

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

      5. unpow2 24.2%

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

      6. hypot-def 57.0%

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

   6. Simplified 57.0%

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

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 7: 45.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B_m}^{2} \leq 10^{-79}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_0\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B_m}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 1e-79)
     (/ (- (sqrt (* (* F t_0) (* 2.0 (+ A A))))) t_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 t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-79) {
		tmp = -sqrt(((F * t_0) * (2.0 * (A + A)))) / t_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)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-79)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + A))))) / t_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_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-79], N[((-N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $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 (pow.f64 B 2) < 1e-79

   1. Initial program 24.0%

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

   3. Simplified 32.4%

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

   4. Taylor expanded in C around inf 15.9%

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

   if 1e-79 < (pow.f64 B 2)

   1. Initial program 15.9%

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

   3. Simplified 16.5%

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

   4. Taylor expanded in C around 0 12.5%

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

      1. associate-*r* 12.5%

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

      2. mul-1-neg 12.5%

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

      3. +-commutative 12.5%

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

      4. unpow2 12.5%

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

      5. unpow2 12.5%

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

      6. hypot-def 24.1%

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

   6. Simplified 24.1%

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

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-79}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 8: 44.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B_m}^{2} \leq 10^{-79}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B_m}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 1e-79)
   (/
    (- (sqrt (* -8.0 (* (* A C) (* F (+ A A))))))
    (fma A (* C -4.0) (pow B_m 2.0)))
   (* (- (/ (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 (pow(B_m, 2.0) <= 1e-79) {
		tmp = -sqrt((-8.0 * ((A * C) * (F * (A + A))))) / fma(A, (C * -4.0), pow(B_m, 2.0));
	} 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 ^ 2.0) <= 1e-79)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A)))))) / fma(A, Float64(C * -4.0), (B_m ^ 2.0)));
	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[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-79], N[((-N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $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 (pow.f64 B 2) < 1e-79

   1. Initial program 24.0%

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

   3. Simplified 25.5%

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

   4. Taylor expanded in C around inf 15.2%

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

      1. associate-*r* 15.9%

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

      2. sub-neg 15.9%

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

      3. mul-1-neg 15.9%

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

      4. remove-double-neg 15.9%

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

   6. Simplified 15.9%

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

   if 1e-79 < (pow.f64 B 2)

   1. Initial program 15.9%

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

   3. Simplified 16.5%

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

   4. Taylor expanded in C around 0 12.5%

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

      1. associate-*r* 12.5%

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

      2. mul-1-neg 12.5%

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

      3. +-commutative 12.5%

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

      4. unpow2 12.5%

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

      5. unpow2 12.5%

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

      6. hypot-def 24.1%

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

   6. Simplified 24.1%

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

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-79}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 9: 30.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\frac{\sqrt{2}}{B_m}\\ \mathbf{if}\;A \leq -2.4 \cdot 10^{+184}:\\ \;\;\;\;t_0 \cdot \sqrt{-0.5 \cdot \frac{{B_m}^{2} \cdot F}{C}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \end{array} \end{array} \]

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 (- (/ (sqrt 2.0) B_m))))
   (if (<= A -2.4e+184)
     (* t_0 (sqrt (* -0.5 (/ (* (pow B_m 2.0) F) C))))
     (* t_0 (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 = -(sqrt(2.0) / B_m);
	double tmp;
	if (A <= -2.4e+184) {
		tmp = t_0 * sqrt((-0.5 * ((pow(B_m, 2.0) * F) / C)));
	} else {
		tmp = t_0 * sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -(Math.sqrt(2.0) / B_m);
	double tmp;
	if (A <= -2.4e+184) {
		tmp = t_0 * Math.sqrt((-0.5 * ((Math.pow(B_m, 2.0) * F) / C)));
	} else {
		tmp = t_0 * Math.sqrt((F * (A - Math.hypot(B_m, A))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -(math.sqrt(2.0) / B_m)
	tmp = 0
	if A <= -2.4e+184:
		tmp = t_0 * math.sqrt((-0.5 * ((math.pow(B_m, 2.0) * F) / C)))
	else:
		tmp = t_0 * math.sqrt((F * (A - math.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 = Float64(-Float64(sqrt(2.0) / B_m))
	tmp = 0.0
	if (A <= -2.4e+184)
		tmp = Float64(t_0 * sqrt(Float64(-0.5 * Float64(Float64((B_m ^ 2.0) * F) / C))));
	else
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(A - hypot(B_m, A)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -(sqrt(2.0) / B_m);
	tmp = 0.0;
	if (A <= -2.4e+184)
		tmp = t_0 * sqrt((-0.5 * (((B_m ^ 2.0) * F) / C)));
	else
		tmp = t_0 * sqrt((F * (A - hypot(B_m, A))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])}, If[LessEqual[A, -2.4e+184], N[(t$95$0 * N[Sqrt[N[(-0.5 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * F), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 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 A < -2.39999999999999997e184

   1. Initial program 1.7%

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

   3. Simplified 7.8%

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

   4. Taylor expanded in A around 0 1.5%

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

      1. associate-*r* 1.5%

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

      2. mul-1-neg 1.5%

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

      3. unpow2 1.5%

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

      4. unpow2 1.5%

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

      5. hypot-def 2.1%

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

   6. Simplified 2.1%

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

   7. Taylor expanded in C around inf 18.6%

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

   if -2.39999999999999997e184 < A

   1. Initial program 22.3%

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

   3. Simplified 22.6%

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

   4. Taylor expanded in C around 0 11.5%

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

      1. associate-*r* 11.5%

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

      2. mul-1-neg 11.5%

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

      3. +-commutative 11.5%

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

      4. unpow2 11.5%

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

      5. unpow2 11.5%

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

      6. hypot-def 18.8%

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

   6. Simplified 18.8%

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

4. Final simplification 18.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.4 \cdot 10^{+184}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{-0.5 \cdot \frac{{B}^{2} \cdot F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 10: 30.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\frac{\sqrt{2}}{B_m}\\ \mathbf{if}\;F \leq -1.9 \cdot 10^{+89}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{C}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ \end{array} \end{array} \]

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 (- (/ (sqrt 2.0) B_m))))
   (if (<= F -1.9e+89)
     (* t_0 (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C)))))
     (* t_0 (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 = -(sqrt(2.0) / B_m);
	double tmp;
	if (F <= -1.9e+89) {
		tmp = t_0 * sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	} else {
		tmp = t_0 * sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -(Math.sqrt(2.0) / B_m);
	double tmp;
	if (F <= -1.9e+89) {
		tmp = t_0 * Math.sqrt((F * (-0.5 * (Math.pow(B_m, 2.0) / C))));
	} else {
		tmp = t_0 * Math.sqrt((F * (A - Math.hypot(B_m, A))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -(math.sqrt(2.0) / B_m)
	tmp = 0
	if F <= -1.9e+89:
		tmp = t_0 * math.sqrt((F * (-0.5 * (math.pow(B_m, 2.0) / C))))
	else:
		tmp = t_0 * math.sqrt((F * (A - math.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 = Float64(-Float64(sqrt(2.0) / B_m))
	tmp = 0.0
	if (F <= -1.9e+89)
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C)))));
	else
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(A - hypot(B_m, A)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -(sqrt(2.0) / B_m);
	tmp = 0.0;
	if (F <= -1.9e+89)
		tmp = t_0 * sqrt((F * (-0.5 * ((B_m ^ 2.0) / C))));
	else
		tmp = t_0 * sqrt((F * (A - hypot(B_m, A))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])}, If[LessEqual[F, -1.9e+89], N[(t$95$0 * N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 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 F < -1.90000000000000012e89

   1. Initial program 15.1%

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

   3. Simplified 11.6%

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

   4. Taylor expanded in A around 0 10.2%

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

      1. associate-*r* 10.2%

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

      2. mul-1-neg 10.2%

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

      3. unpow2 10.2%

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

      4. unpow2 10.2%

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

      5. hypot-def 10.3%

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

   6. Simplified 10.3%

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

   7. Taylor expanded in C around inf 5.9%

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

   if -1.90000000000000012e89 < F

   1. Initial program 21.7%

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

   3. Simplified 24.5%

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

   4. Taylor expanded in C around 0 9.6%

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

      1. associate-*r* 9.6%

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

      2. mul-1-neg 9.6%

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

      3. +-commutative 9.6%

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

      4. unpow2 9.6%

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

      5. unpow2 9.6%

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

      6. hypot-def 18.5%

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

   6. Simplified 18.5%

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

4. Final simplification 14.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{+89}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{C}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 11: 31.0% accurate, 2.0× speedup

Math

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

FPCore

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[((-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. Initial program 19.8%

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

3. Simplified 20.8%

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

4. Taylor expanded in C around 0 10.1%

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

   1. associate-*r* 10.1%

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

   2. mul-1-neg 10.1%

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

   3. +-commutative 10.1%

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

   4. unpow2 10.1%

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

   5. unpow2 10.1%

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

   6. hypot-def 16.5%

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

6. Simplified 16.5%

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

7. Final simplification 16.5%

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

Alternative 12: 25.8% accurate, 3.0× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Simplified 20.8%

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

4. Taylor expanded in A around 0 9.3%

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

   1. associate-*r* 9.3%

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

   2. mul-1-neg 9.3%

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

   3. unpow2 9.3%

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

   4. unpow2 9.3%

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

   5. hypot-def 15.8%

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

6. Simplified 15.8%

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

7. Taylor expanded in C around 0 15.3%

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

   1. mul-1-neg 15.3%

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

9. Simplified 15.3%

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

10. Final simplification 15.3%

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

Alternative 13: 0.6% accurate, 3.0× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Simplified 20.8%

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

4. Taylor expanded in A around 0 9.3%

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

   1. associate-*r* 9.3%

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

   2. mul-1-neg 9.3%

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

   3. unpow2 9.3%

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

   4. unpow2 9.3%

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

   5. hypot-def 15.8%

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

6. Simplified 15.8%

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

7. Taylor expanded in B around -inf 1.6%

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

8. Final simplification 1.6%

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

Alternative 14: 0.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Simplified 25.9%

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

   1. add-sqr-sqrt 25.8%

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

   2. pow2 25.8%

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

   3. pow1/2 25.8%

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

   4. sqrt-pow1 25.8%

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

   5. associate-*l* 26.4%

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

   6. associate-+r- 25.8%

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

   7. metadata-eval 25.8%

      \[\leadsto \frac{-{\left({\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

5. Applied egg-rr 25.8%

   \[\leadsto \frac{-\color{blue}{{\left({\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

6. Taylor expanded in A around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

8. Simplified 0.0%

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

9. Final simplification 0.0%

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

Reproduce

herbie shell --seed 2023321 
(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))))